Journal of Differential Geometry Articles (Project Euclid)
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The latest articles from Journal of Differential Geometry on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 04 May 2011 09:16 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Classification of compact ancient solutions to the curve shortening flow
http://projecteuclid.org/euclid.jdg/1279114297
<strong>Panagiota Daskalopoulos</strong>, <strong>Richard Hamilton</strong>, <strong>Natasa Sesum</strong><p><strong>Source: </strong>J. Differential Geom., Volume 84, Number 3, 455--464.</p>projecteuclid.org/euclid.jdg/1279114297_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTSubspace concentration of dual curvature measures of symmetric convex bodieshttps://projecteuclid.org/euclid.jdg/1531188189<strong>Károly J. Böröczky</strong>, <strong>Martin Henk</strong>, <strong>Hannes Pollehn</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 411--429.</p><p><strong>Abstract:</strong><br/>
We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body.
</p>projecteuclid.org/euclid.jdg/1531188189_20180709220319Mon, 09 Jul 2018 22:03 EDTA discrete uniformization theorem for polyhedral surfaces IIhttps://projecteuclid.org/euclid.jdg/1531188190<strong>Xianfeng Gu</strong>, <strong>Ren Guo</strong>, <strong>Feng Luo</strong>, <strong>Jian Sun</strong>, <strong>Tianqi Wu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 431--466.</p><p><strong>Abstract:</strong><br/>
A notion of discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss–Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.
</p>projecteuclid.org/euclid.jdg/1531188190_20180709220319Mon, 09 Jul 2018 22:03 EDTLooijenga’s conjecture via integral-affine geometryhttps://projecteuclid.org/euclid.jdg/1531188193<strong>Philip Engel</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 467--495.</p><p><strong>Abstract:</strong><br/>
A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. Recent work of Gross, Hacking, and Keel has proven Looijenga’s conjecture using methods from mirror symmetry. This paper provides an alternative proof of Looijenga’s conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983.
</p>projecteuclid.org/euclid.jdg/1531188193_20180709220319Mon, 09 Jul 2018 22:03 EDTThe intersection of a hyperplane with a lightcone in the Minkowski spacetimehttps://projecteuclid.org/euclid.jdg/1531188194<strong>Pengyu Le</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 497--507.</p><p><strong>Abstract:</strong><br/>
Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a geometric interpretation of the Green’s function of the Laplacian on the standard sphere. In the end, we contribute a clearer understanding of the anisotropic criterion for formation of trapped surfaces in vacuum.
</p>projecteuclid.org/euclid.jdg/1531188194_20180709220319Mon, 09 Jul 2018 22:03 EDTThe local picture theorem on the scale of topologyhttps://projecteuclid.org/euclid.jdg/1531188195<strong>William H. Meeks</strong>, <strong>Joaquín Pérez</strong>, <strong>Antonio Ros</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3, 509--565.</p><p><strong>Abstract:</strong><br/>
We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius . This structure theorem includes a limit object which we call a minimal parking garage structure on $\mathbb{R}^3$, whose theory we also develop.
</p>projecteuclid.org/euclid.jdg/1531188195_20180709220319Mon, 09 Jul 2018 22:03 EDTIndex to Volume 109https://projecteuclid.org/euclid.jdg/1531188196<p><strong>Source: </strong>Journal of Differential Geometry, Volume 109, Number 3</p>projecteuclid.org/euclid.jdg/1531188196_20180709220319Mon, 09 Jul 2018 22:03 EDTThe $L_p$-Aleksandrov problem for $L_p$-integral curvaturehttps://projecteuclid.org/euclid.jdg/1536285625<strong>Yong Huang</strong>, <strong>Erwin Lutwak</strong>, <strong>Deane Yang</strong>, <strong>Gaoyong Zhang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 1--29.</p><p><strong>Abstract:</strong><br/>
It is shown that within the $L_p$-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural $L_p$ extension, for all real $p$. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the $L_p$-integral curvature of a convex body. This problem is solved for positive $p$ and is answered for negative $p$ provided the given measure is even.
</p>projecteuclid.org/euclid.jdg/1536285625_20180906220045Thu, 06 Sep 2018 22:00 EDTEntropy of closed surfaces and min-max theoryhttps://projecteuclid.org/euclid.jdg/1536285626<strong>Daniel Ketover</strong>, <strong>Xin Zhou</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 31--71.</p><p><strong>Abstract:</strong><br/>
Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding–Ilmanen–Minicozzi–White conjectured (since proved by Bernstein–Wang) that the entropy of any closed surface is at least that of the self-shrinking two-sphere. In this paper we give an alternative proof of their conjecture for closed embedded 2-spheres. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques–Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in $\mathbb{R}^3$ which we also establish. To any closed surface in $\mathbb{R}^3$ we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for $\mathbb{R}^3$ with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of nonstationary varifolds (with respect to the Gaussian metric of $\mathbb{R}^3$) in a continuous manner.
</p>projecteuclid.org/euclid.jdg/1536285626_20180906220045Thu, 06 Sep 2018 22:00 EDTOn the local extension of the future null infinityhttps://projecteuclid.org/euclid.jdg/1536285627<strong>Junbin Li</strong>, <strong>Xi-Ping Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 73--133.</p><p><strong>Abstract:</strong><br/>
We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].
</p>projecteuclid.org/euclid.jdg/1536285627_20180906220045Thu, 06 Sep 2018 22:00 EDTWeakly pseudoconvex Kähler manifoldshttps://projecteuclid.org/euclid.jdg/1536285628<strong>Xiangyu Zhou</strong>, <strong>Langfeng Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 1, 135--186.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove an $L^2$ extension theorem for holomorphic sections of holomorphic line bundles equipped with singular metrics on weakly pseudoconvex Kähler manifolds. Furthermore, in our $L^2$ estimate, optimal constants corresponding to variable denominators are obtained. As applications, we prove an $L^q$ extension theorem with an optimal estimate on weakly pseudoconvex Kähler manifolds and the log-plurisubharmonicity of the fiberwise Bergman kernel in the Kähler case.
</p>projecteuclid.org/euclid.jdg/1536285628_20180906220045Thu, 06 Sep 2018 22:00 EDTThe floating body in real space formshttps://projecteuclid.org/euclid.jdg/1538791243<strong>Florian Besau</strong>, <strong>Elisabeth M. Werner</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 187--220.</p><p><strong>Abstract:</strong><br/>
We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space.
Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.
</p>projecteuclid.org/euclid.jdg/1538791243_20181005220130Fri, 05 Oct 2018 22:01 EDTAsymptotics for the wave equation on differential forms on Kerr–de Sitter spacehttps://projecteuclid.org/euclid.jdg/1538791244<strong>Peter Hintz</strong>, <strong>András Vasy</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 221--279.</p><p><strong>Abstract:</strong><br/>
We study asymptotics for solutions of Maxwell’s equations, in fact, of the Hodge–de Rham equation $(d+\delta)u = 0$ without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which, in particular, include Schwarzschild—de Sitter spaces of all spacetime dimensions $n \geq 4$. We prove that solutions decay exponentially to $0$ or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and, in particular, prove analogous results on Schwarzschild–de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell’s equations, the Hodge–de Rham equation and the wave equation on differential forms on Kerr–de Sitter spacetimes with small angular momentum.
</p>projecteuclid.org/euclid.jdg/1538791244_20181005220130Fri, 05 Oct 2018 22:01 EDTNaturality of Heegaard Floer invariants under positive rational contact surgeryhttps://projecteuclid.org/euclid.jdg/1538791245<strong>Thomas E. Mark</strong>, <strong>Bülent Tosun</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 281--344.</p><p><strong>Abstract:</strong><br/>
For a nullhomologous Legendrian knot in a closed contact $3$-manifold $Y$ we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of $Y$ is mapped by a surgery cobordism to the contact invariant of the result of contact surgery, and we characterize the $\mathrm{spin}^c$ structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery on a Legendrian knot in the standard $3$-sphere, generalizing previous results of Lisca–Stipsicz and Golla. In fact, our methods allow direct calculation of the contact invariant in terms of the rational surgery mapping cone of Ozsváth and Szabó. The proof involves a construction called reducible open book surgery, which reduces in special cases to the capping-off construction studied by Baldwin.
</p>projecteuclid.org/euclid.jdg/1538791245_20181005220130Fri, 05 Oct 2018 22:01 EDTEmbeddedness of least area minimal hypersurfaceshttps://projecteuclid.org/euclid.jdg/1538791246<strong>Antoine Song</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 2, 345--377.</p><p><strong>Abstract:</strong><br/>
In “Simple closed geodesics on convex surfaces” [ J. Differential Geom. , 36(3):517–549, 1992], E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n \leq 6$, a least area closed minimal hypersurface exists and any such hypersurface is embedded.
As an application, we give a short proof of the fact that if a closed three-manifold $M$ has scalar curvature at least $6$ and is not isometric to the round three-sphere, then $M$ contains an embedded closed minimal surface of area less than $4 \pi$. This confirms a conjecture of F. C. Marques and A. Neves.
</p>projecteuclid.org/euclid.jdg/1538791246_20181005220130Fri, 05 Oct 2018 22:01 EDTNon-convex balls in the Teichmüller metrichttps://projecteuclid.org/euclid.jdg/1542423625<strong>Maxime Fortier Bourque</strong>, <strong>Kasra Rafi</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 379--412.</p><p><strong>Abstract:</strong><br/>
We prove that the Teichmüller space of surfaces of genus $g$ with $p$ punctures contains balls which are not convex in the Teichmüller metric whenever its complex dimension $(3g −3+p)$ is greater than $1$.
</p>projecteuclid.org/euclid.jdg/1542423625_20181116220051Fri, 16 Nov 2018 22:00 ESTNavigating the space of symmetric CMC surfaceshttps://projecteuclid.org/euclid.jdg/1542423626<strong>Lynn Heller</strong>, <strong>Sebastian Heller</strong>, <strong>Nicholas Schmitt</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 413--455.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. By construction the flow yields closed (possibly branched) CMC surfaces at rational times and immersed higher genus CMC surfaces at integer times. We prove the short time existence of this flow near the spectral data of (certain classes of) CMC tori and obtain thereby the existence of new families of closed (possibly branched) connected CMC surfaces of higher genus. Moreover, we prove that flowing the spectral data for the Clifford torus is equivalent to the flow of Plateau solutions by varying the angle of the fundamental piece in Lawson’s construction for the minimal surfaces $\xi_{g,1}$.
</p>projecteuclid.org/euclid.jdg/1542423626_20181116220051Fri, 16 Nov 2018 22:00 ESTInverse problems for the connection Laplacianhttps://projecteuclid.org/euclid.jdg/1542423627<strong>Yaroslav Kurylev</strong>, <strong>Lauri Oksanen</strong>, <strong>Gabriel P. Paternain</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 457--494.</p><p><strong>Abstract:</strong><br/>
We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a Calderón problem for connections on a cylinder.
</p>projecteuclid.org/euclid.jdg/1542423627_20181116220051Fri, 16 Nov 2018 22:00 ESTTowards $A+B$ theory in conifold transitions for Calabi–Yau threefoldshttps://projecteuclid.org/euclid.jdg/1542423628<strong>Yuan-Pin Lee</strong>, <strong>Hui-Wen Lin</strong>, <strong>Chin-Lung Wang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 495--541.</p><p><strong>Abstract:</strong><br/>
For projective conifold transitions between Calabi–Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov–Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.
</p>projecteuclid.org/euclid.jdg/1542423628_20181116220051Fri, 16 Nov 2018 22:00 ESTExistence of solutions to the even dual Minkowski problemhttps://projecteuclid.org/euclid.jdg/1542423629<strong>Yiming Zhao</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3, 543--572.</p><p><strong>Abstract:</strong><br/>
Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn–Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is $0$) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space)—two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by Böröczky, Henk & Pollehn that these new sufficiency conditions are also necessary.
</p>projecteuclid.org/euclid.jdg/1542423629_20181116220051Fri, 16 Nov 2018 22:00 ESTIndex to Volume 110https://projecteuclid.org/euclid.jdg/1543287636<p><strong>Source: </strong>Journal of Differential Geometry, Volume 110, Number 3</p>projecteuclid.org/euclid.jdg/1543287636_20181126220119Mon, 26 Nov 2018 22:01 ESTEinstein solvmanifolds have maximal symmetryhttps://projecteuclid.org/euclid.jdg/1547607686<strong>Carolyn S. Gordon</strong>, <strong>Michael R. Jablonski</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 1--38.</p><p><strong>Abstract:</strong><br/>
All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ricci curvature on a solvable Lie group is maximally symmetric. This theorem is motivated both by the Alekseevskii Conjecture and by the question of stability of Einstein metrics under the Ricci flow. We also address questions of existence of maximally symmetric left-invariant Riemannian metrics more generally.
</p>projecteuclid.org/euclid.jdg/1547607686_20190115220152Tue, 15 Jan 2019 22:01 ESTOn short time existence for the planar network flowhttps://projecteuclid.org/euclid.jdg/1547607687<strong>Tom Ilmanen</strong>, <strong>André Neves</strong>, <strong>Felix Schulze</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 39--89.</p><p><strong>Abstract:</strong><br/>
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White’s local regularity theorem for mean curvature flow. We also show a pseudolocality theorem for mean curvature flow in any codimension, assuming only that the initial submanifold can be locally written as a graph with sufficiently small Lipschitz constant.
</p>projecteuclid.org/euclid.jdg/1547607687_20190115220152Tue, 15 Jan 2019 22:01 ESTVanishing Pohozaev constant and removability of singularitieshttps://projecteuclid.org/euclid.jdg/1547607688<strong>Jürgen Jost</strong>, <strong>Chunqin Zhou</strong>, <strong>Miaomiao Zhu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 91--144.</p><p><strong>Abstract:</strong><br/>
Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the validity of the Pohozaev identity. In situations where such an identity fails to hold, we introduce a new quantity, called the Pohozaev constant , which on one hand measures the extent to which the Pohozaev identity fails and, on the other hand, provides a characterization of the singular behavior of a solution at an isolated singularity. We apply this to the blow-up analysis for super-Liouville type equations on Riemann surfaces with conical singularities, because in the presence of such singularities, conformal invariance no longer holds and a local singularity is in general non-removable unless the Pohozaev constant is vanishing.
</p>projecteuclid.org/euclid.jdg/1547607688_20190115220152Tue, 15 Jan 2019 22:01 EST$n$-dimension central affine curve flowshttps://projecteuclid.org/euclid.jdg/1547607689<strong>Chuu-Lian Terng</strong>, <strong>Zhiwei Wu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 1, 145--189.</p><p><strong>Abstract:</strong><br/>
For $n$-dimensional central affine curve flows, we
1) solve the Cauchy problem with periodic initial data and with initial data having rapidly decaying central affine curvatures,
2) construct Bäcklund transformations, a Permutability formula, and explicit solutions,
3) write down formulas for the Bi-Hamiltonian structure and conservation laws.
</p>projecteuclid.org/euclid.jdg/1547607689_20190115220152Tue, 15 Jan 2019 22:01 ESTNull mean curvature flow and outermost MOTShttps://projecteuclid.org/euclid.jdg/1549422101<strong>Theodora Bourni</strong>, <strong>Kristen Moore</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 191--239.</p><p><strong>Abstract:</strong><br/>
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partial \Omega$, we show that there exists a weak solution to the null mean curvature flow, given as a limit of approximate solutions that are defined using the $\varepsilon$-regularization method. We show that the approximate solutions blow up on the outermost MOTS and the weak solution converges (as boundaries of finite perimeter sets) to a generalized MOTS.
</p>projecteuclid.org/euclid.jdg/1549422101_20190205220216Tue, 05 Feb 2019 22:02 ESTFaltings delta-invariant and semistable degenerationhttps://projecteuclid.org/euclid.jdg/1549422102<strong>Robin de Jong</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 241--301.</p><p><strong>Abstract:</strong><br/>
We determine the asymptotic behavior of the Arakelov metric, the Arakelov–Green’s function, and the Faltings delta-invariant for arbitrary one-parameter families of complex curves with semistable degeneration. The leading terms in the asymptotics are given a combinatorial interpretation in terms of S. Zhang’s theory of admissible Green’s functions on polarized metrized graphs.
</p>projecteuclid.org/euclid.jdg/1549422102_20190205220216Tue, 05 Feb 2019 22:02 ESTQuasi-negative holomorphic sectional curvature and positivity of the canonical bundlehttps://projecteuclid.org/euclid.jdg/1549422103<strong>Simone Diverio</strong>, <strong>Stefano Trapani</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 303--314.</p><p><strong>Abstract:</strong><br/>
We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non-positive and strictly negative in at least one point, then its canonical bundle is positive. This answers in the affirmative to a question first asked by S.-T. Yau.
</p>projecteuclid.org/euclid.jdg/1549422103_20190205220216Tue, 05 Feb 2019 22:02 ESTOn positive scalar curvature and moduli of curveshttps://projecteuclid.org/euclid.jdg/1549422104<strong>Kefeng Liu</strong>, <strong>Yunhui Wu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 315--338.</p><p><strong>Abstract:</strong><br/>
In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g \geqslant 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that ${\lVert \, \cdotp \rVert}_{ds^2} \succ {\lVert \, \cdotp \rVert}_T$ where ${\lVert \, \cdotp \rVert}_T$ is the Teichmüller metric.
Our second result is the proof that any cover $M$ of the moduli space $\mathbb{M}_g$ of a closed Riemann surface $S_g$ does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichmüller metric, which implies a conjecture of Farb–Weinberger in [9].
</p>projecteuclid.org/euclid.jdg/1549422104_20190205220216Tue, 05 Feb 2019 22:02 ESTA sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operatorshttps://projecteuclid.org/euclid.jdg/1549422105<strong>Dario Prandi</strong>, <strong>Luca Rizzi</strong>, <strong>Marcello Seri</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 2, 339--379.</p><p><strong>Abstract:</strong><br/>
In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g., CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a “reduction procedure” that allows to consider only a simple subset of sub-Riemannian geodesics.
As an application, we derive isoperimetric-type and ($p$-)Hardy-type inequalities for a compact domain $M$ with piecewise $C^{1,1}$ boundary, and a universal lower bound for the first Dirichlet eigenvalue $\lambda_1 (M)$ of the sub-Laplacian,
\[ \lambda_1 (M) \geq \frac{k \pi^2}{L^2} \; \textrm{,} \]
in terms of the rank $k$ of the distribution and the length $L$ of the longest reduced sub-Riemannian geodesic contained in $M$. All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations:
\[ \mathbb{S}^1 \hookrightarrow \mathbb{S}^{2d+1} \overset{p}{\to} \mathbb{CP}^d \; \textrm{,} \qquad \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3} \overset{p}{\to} \mathbb{HP}^d \; \textrm{,} \qquad {d \geq 1} \; \textrm{,} \]
where the sub-Laplacian is the standard hypoelliptic operator of CR and QC geometries, $L = \pi$ and $k = 2d$ or $4d$, respectively.
</p>projecteuclid.org/euclid.jdg/1549422105_20190205220216Tue, 05 Feb 2019 22:02 ESTUnique asymptotics of ancient convex mean curvature flow solutionshttps://projecteuclid.org/euclid.jdg/1552442605<strong>Sigurd Angenent</strong>, <strong>Panagiota Daskalopoulos</strong>, <strong>Natasa Sesum</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 381--455.</p><p><strong>Abstract:</strong><br/>
We study compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1) \times O(n)$ symmetry. We show they all have unique asymptotics as $t \to -\infty$ and we give a precise asymptotic description of these solutions. The asymptotics apply, in particular, to the solutions constructed by White, and Haslhofer and Hershkovits (in the case of those particular solutions the asymptotics were predicted and formally computed by Angenent).
</p>projecteuclid.org/euclid.jdg/1552442605_20190312220339Tue, 12 Mar 2019 22:03 EDTImmersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic $3$-manifoldshttps://projecteuclid.org/euclid.jdg/1552442607<strong>Yi Liu</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 457--493.</p><p><strong>Abstract:</strong><br/>
In this paper, it is shown that every closed hyperbolic $3$-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two frames with a path of frames in a prescribed relative homology class of the frame bundle. The existence result is applied to show that every uniform lattice of $\mathrm{PSL}(2, \mathbb{C})$ admits an exhausting nested sequence of sublattices with exponential homological torsion growth. However, the constructed sublattices are not normal in general.
</p>projecteuclid.org/euclid.jdg/1552442607_20190312220339Tue, 12 Mar 2019 22:03 EDTStability of torsion-free $\mathrm{G}_2$ structures along the Laplacian flowhttps://projecteuclid.org/euclid.jdg/1552442608<strong>Jason D. Lotay</strong>, <strong>Yong Wei</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 495--526.</p><p><strong>Abstract:</strong><br/>
We prove that torsion-free $\mathrm{G}_2$ structures are (weakly) dynamically stable along the Laplacian flow for closed $\mathrm{G}_2$ structures. More precisely, given a torsion-free $\mathrm{G}_2$ structure $\overline{\varphi}$ on a compact $7$-manifold $M$, the Laplacian flow with initial value in $[\overline{\varphi}]$, sufficiently close to $\overline{\varphi}$, will converge to a point in the $\mathrm{Diff}^0 (M)$-orbit of $\overline{\varphi}$. We deduce, from fundamental work of Joyce, that the Laplacian flow starting at any closed $\mathrm{G}_2$ structure with sufficiently small torsion will exist for all time and converge to a torsion-free $\mathrm{G}_2$ structure.
</p>projecteuclid.org/euclid.jdg/1552442608_20190312220339Tue, 12 Mar 2019 22:03 EDTDecorated super-Teichmüller spacehttps://projecteuclid.org/euclid.jdg/1552442609<strong>R. C. Penner</strong>, <strong>Anton M. Zeitlin</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 527--566.</p><p><strong>Abstract:</strong><br/>
We introduce coordinates for a principal bundle $S\tilde{T}(F)$ over the super Teichmüller space $ST(F)$ of a surface F with $s \geq 1$ punctures that extend the lambda length coordinates on the decorated bundle $\tilde{T}(F) = T(F) \times \mathbb{R}^s_{+}$ over the usual Teichmüller space $T(F)$. In effect, the action of a Fuchsian subgroup of $PSL (2, \mathbb{R})$ on Minkowski space $\mathbb{R}^{2,1}$ is replaced by the action of a super Fuchsian subgroup of $OSp (1\vert 2)$ on the super Minkowski space $\mathbb{R}^{2, 1 \vert 2}$, where $OSp (1\vert 2)$ denotes the orthosymplectic Lie supergroup, and the lambda lengths are extended by fermionic invariants of suitable triples of isotropic vectors in $\mathbb{R}^{2, 1 \vert 2}$. As in the bosonic case, there is the analogue of the Ptolemy transformation now on both even and odd coordinates as well as an invariant even two-form on $S\tilde{T}(F)$ generalizing the Weil–Petersson Kähler form. This, finally, solves a problem posed in Yuri Ivanovitch Manin’s Moscow seminar some thirty years ago to find the super analogue of decorated Teichmüller theory and provides a natural geometric interpretation in $\mathbb{R}^{2, 1 \vert 2}$ for the super moduli of $S\tilde{T}(F)$.
</p>projecteuclid.org/euclid.jdg/1552442609_20190312220339Tue, 12 Mar 2019 22:03 EDTKohn–Rossi cohomology and nonexistence of CR morphisms between compact strongly pseudoconvex CR manifoldshttps://projecteuclid.org/euclid.jdg/1552442610<strong>Stephen S.-T. Yau</strong>, <strong>Huaiqing Zuo</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3, 567--580.</p><p><strong>Abstract:</strong><br/>
One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds $X_1$ and $X_2$ of dimension $2n-1$, is there a non-constant CR morphism between them? In this paper, we use Kohn–Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specifically, if $\dim H^{p,q}_{KR} (X_1) \lt \dim H^{p,q}_{KR} (X_2)$ for any $(p, q)$ with $1 \leq q \leq n-2$, then there is no non-constant CR morphism from $X_1$ to $X_2$.
</p>projecteuclid.org/euclid.jdg/1552442610_20190312220339Tue, 12 Mar 2019 22:03 EDTIndex to Volume 111https://projecteuclid.org/euclid.jdg/1552442611<p><strong>Source: </strong>Journal of Differential Geometry, Volume 111, Number 3</p>projecteuclid.org/euclid.jdg/1552442611_20190312220339Tue, 12 Mar 2019 22:03 EDTLagrangian cobordism and metric invariantshttps://projecteuclid.org/euclid.jdg/1557281005<strong>Octav Cornea</strong>, <strong>Egor Shelukhin</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 1--45.</p><p><strong>Abstract:</strong><br/>
We introduce new pseudo-metrics on spaces of Lagrangian submanifolds of a symplectic manifold $(M, \omega)$ by considering areas associated to projecting Lagrangian cobordisms in $\mathbb{C} \times M$ to the “time-energy plane” $\mathbb{C}$. We investigate the non-degeneracy properties of these pseudo-metrics, reflecting the rigidity and flexibility aspects of Lagrangian cobordisms.
</p>projecteuclid.org/euclid.jdg/1557281005_20190507220348Tue, 07 May 2019 22:03 EDTMinimal surfaces for Hitchin representationshttps://projecteuclid.org/euclid.jdg/1557281006<strong>Song Dai</strong>, <strong>Qiongling Li</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 47--77.</p><p><strong>Abstract:</strong><br/>
Given a reductive representation $\rho : \pi_1 (S) \to G$, there exists a $\rho$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $\Sigma$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: the $q_n$ and $q_{n-1}$ cases. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system.
</p>projecteuclid.org/euclid.jdg/1557281006_20190507220348Tue, 07 May 2019 22:03 EDTALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surfacehttps://projecteuclid.org/euclid.jdg/1557281007<strong>Lorenzo Foscolo</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 79--120.</p><p><strong>Abstract:</strong><br/>
We construct large families of new collapsing hyperkähler metrics on the $K3$ surface. The limit space is a flat Riemannian $3$-orbifold $T^3 / \mathbb{Z}_2$. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most $24$ exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on $T^3$. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type ($D_k$) for the fixed points of the involution on T3 and of cyclic type ($A_k$) otherwise.
The collapsing metrics are constructed by deforming approximately hyperkähler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) $S^1$–invariant hyperkähler metric arising from the Gibbons–Hawking ansatz over a punctured $3$-torus.
As an immediate application to submanifold geometry, we exhibit hyperkähler metrics on the $K3$ surface that admit a strictly stable minimal sphere which cannot be holomorphic with respect to any complex structure compatible with the metric.
</p>projecteuclid.org/euclid.jdg/1557281007_20190507220348Tue, 07 May 2019 22:03 EDTReal submanifolds of maximum complex tangent space at a CR singular point, IIhttps://projecteuclid.org/euclid.jdg/1557281008<strong>Xianghong Gong</strong>, <strong>Laurent Stolovitch</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 1, 121--198.</p><p><strong>Abstract:</strong><br/>
We study germs of real analytic $n$-dimensional submanifold of $\mathbf{C}^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions, we first classify holomorphically the quadrics having this property. We then study higher order perturbations of these quadrics and their transformations to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We are led to study formal Poincaré–Dulac normal forms (non-unique) of reversible biholomorphisms. We exhibit a reversible map of which the normal forms are all divergent at the singularity. We then construct a unique formal normal form of the submanifolds under a non degeneracy condition.
</p>projecteuclid.org/euclid.jdg/1557281008_20190507220348Tue, 07 May 2019 22:03 EDTSymplectic embeddings from concave toric domains into convex oneshttps://projecteuclid.org/euclid.jdg/1559786421<strong>Dan Cristofaro-Gardiner</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 199--232.</p><p><strong>Abstract:</strong><br/>
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In “Symplectic embeddings into four-dimensional concave toric domains”, the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all “concave toric domains”, and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into “convex” ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which shows that they are determined by the ECH capacities of a corresponding collection of balls.
</p>projecteuclid.org/euclid.jdg/1559786421_20190605220045Wed, 05 Jun 2019 22:00 EDTProperly immersed surfaces in hyperbolic $3$-manifoldshttps://projecteuclid.org/euclid.jdg/1559786424<strong>William H. Meeks</strong>, <strong>Álvaro K. Ramos</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 233--261.</p><p><strong>Abstract:</strong><br/>
We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N \leq -a^2 \leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $\Sigma$ must be proper in $N$ and its total curvature must be equal to $2 \pi \chi (\Sigma)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than $1$, then we prove that each end of $\Sigma$ is asymptotic (with finite positive integer multiplicity) to a totally umbilic annulus, properly embedded in $N$.
</p>projecteuclid.org/euclid.jdg/1559786424_20190605220045Wed, 05 Jun 2019 22:00 EDTRigidity of pairs of rational homogeneous spaces of Picard number $1$ and analytic continuation of geometric substructures on uniruled projective manifoldshttps://projecteuclid.org/euclid.jdg/1559786425<strong>Ngaiming Mok</strong>, <strong>Yunxin Zhang</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 263--345.</p><p><strong>Abstract:</strong><br/>
Building on the geometric theory of uniruled projective manifolds by Hwang–Mok, which relies on the study of varieties of minimal rational tangents (VMRTs) from both the algebro-geometric and the differential-geometric perspectives, Mok, Hong–Mok and Hong–Park have studied standard embeddings between rational homogeneous spaces $X = G/P$ of Picard number $1$. Denoting by $S \subset X$ an arbitrary germ of complex submanifold which inherits from $X$ a geometric structure defined by taking intersections of VMRTs with tangent subspaces and modeled on some rational homogeneous space $X_0 = G_0 / P_0$ of Picard number $1$ embedded in $X = G/P$ as a linear section through a standard embedding, we say that $(X_0, X)$ is rigid if there always exists some $\gamma \in \mathrm{Aut}(X)$ such that $S$ is an open subset of $\gamma (X_0)$. We prove that a pair $(X_0, X)$ of sub-diagram type is rigid whenever $X_0$ is nonlinear, which in the Hermitian symmetric case recovers Schubert rigidity for nonlinear smooth Schubert cycles, and which in the general rational homogeneous case goes beyond earlier works dealing with images of holomorphic maps. Our methods apply to uniruled projective manifolds $(X, \mathcal{K})$, for which we introduce a general notion of sub-VMRT structures $\varpi : \mathscr{C} (S) \to S$, proving that they are rationally saturated under an auxiliary condition on the intersection $\mathscr{C} (S) := \mathscr{C} (X) \cap \mathbb{P} T (S)$ and a nondegeneracy condition for substructures expressed in terms of second fundamental forms on VMRTs. Under the additional hypothesis that minimal rational curves are of degree $1$ and that distributions spanned by sub-VMRTs are bracket generating, we prove that $S$ extends to a subvariety $Z \subset X$. For its proof, starting with a “Thickening Lemma” which yields smooth collars around certain standard rational curves, we show that the germ of submanifold $(S; x_0)$ and, hence, the associated germ of sub-VMRT structure on $(S; x_0)$ can be propagated along chains of “thickening” curves issuing from $x_0$, and construct by analytic continuation a projective family of chains of rational curves compactifying the latter family, thereby constructing the projective completion $Z$ of $S$ as its image under the evaluation map.
</p>projecteuclid.org/euclid.jdg/1559786425_20190605220045Wed, 05 Jun 2019 22:00 EDTSharp fundamental gap estimate on convex domains of spherehttps://projecteuclid.org/euclid.jdg/1559786428<strong>Shoo Seto</strong>, <strong>Lili Wang</strong>, <strong>Guofang Wei</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 2, 347--389.</p><p><strong>Abstract:</strong><br/>
In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space [3] and conjectured similar results hold for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely when $D$, the diameter of a convex domain in the unit $\mathbb{S}^n$ sphere, is $\leq \frac{\pi}{2}$, the gap is greater than the gap of the corresponding 1-dim sphere model. We also prove the gap is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$, giving a sharp bound. As in [3], the key is to prove a super log-concavity of the first eigenfunction.
</p>projecteuclid.org/euclid.jdg/1559786428_20190605220045Wed, 05 Jun 2019 22:00 EDTDehn filling and the Thurston normhttps://projecteuclid.org/euclid.jdg/1563242469<strong>Kenneth L. Baker</strong>, <strong>Scott A. Taylor</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 391--409.</p><p><strong>Abstract:</strong><br/>
For a compact, orientable, irreducible $3$-manifold with toroidal boundary that is not the product of a torus and an interval or a cable space, each boundary torus has a finite set of slopes such that, if avoided, the Thurston norm of a Dehn filling behaves predictably. More precisely, for all but finitely many slopes, the Thurston norm of a class in the second homology of the filled manifold plus the so-called winding norm of the class will be equal to the Thurston norm of the corresponding class in the second homology of the unfilled manifold. This generalizes a result of Sela and is used to answer a question of Baker–Motegi concerning the Seifert genus of knots obtained by twisting a given initial knot along an unknot which links it.
</p>projecteuclid.org/euclid.jdg/1563242469_20190715220128Mon, 15 Jul 2019 22:01 EDTMin-max embedded geodesic lines in asymptotically conical surfaceshttps://projecteuclid.org/euclid.jdg/1563242470<strong>Alessandro Carlotto</strong>, <strong>Camillo De Lellis</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 411--445.</p><p><strong>Abstract:</strong><br/>
We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we produce, which will be always less or equal than one (with equality under suitable curvature or genericity assumptions), as well as of their precise asymptotic behavior. In fact, we can prove that in any such surface for every couple of opposite half-lines there exists an embedded geodesic line whose two ends are asymptotic, in a suitable sense, to those half-lines.
</p>projecteuclid.org/euclid.jdg/1563242470_20190715220128Mon, 15 Jul 2019 22:01 EDTQuantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaceshttps://projecteuclid.org/euclid.jdg/1563242471<strong>Eleonora Cinti</strong>, <strong>Joaquim Serra</strong>, <strong>Enrico Valdinoci</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 447--504.</p><p><strong>Abstract:</strong><br/>
We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case.
On the one hand, we establish universal $BV$-estimates in every dimension $n \geqslant 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $\mathbb{R}^3$.
On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n = 2, 3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ – with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.
</p>projecteuclid.org/euclid.jdg/1563242471_20190715220128Mon, 15 Jul 2019 22:01 EDTLorentzian Einstein metrics with prescribed conformal infinityhttps://projecteuclid.org/euclid.jdg/1563242472<strong>Alberto Enciso</strong>, <strong>Niky Kamran</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 505--554.</p><p><strong>Abstract:</strong><br/>
We prove a local well-posedness theorem for the $(n+1)$-dimensional Einstein equations in Lorentzian signature, with initial data $(\widetilde{g},K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\widehat{g}$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an $n$-dimensional asymptotically hyperbolic Riemannian manifold $(M, \widetilde{g})$ such that the conformally rescaled metric $x^2 \widetilde{g}$ (with $x$ a boundary defining function) extends to the closure $\overline{M}$ of $M$ as a metric of class $C^{n-1} (\overline{M})$ which is also poly-homogeneous of class $C^p_{\mathrm{polyhom}} (\overline{M})$. Likewise we assume that the conformally rescaled symmetric $(0, 2)$-tensor $x^ 2 K$ extends to $\overline{M}$ as a tensor field of class $C^{n-1} (\overline{M})$ which is polyhomogeneous of class $C^{p-1}_{\mathrm{polyhom}} (\overline{M})$. We assume that the initial data $(\widetilde{g}, K)$ satisfy the Einstein constraint equations and also that the boundary datum is of class $C^p$ on $\partial M \times (-T_0, T_0)$ and satisfies a set of natural compatibility conditions with the initial data. We then prove that there exists an integer $r_n$, depending only on the dimension $n$, such that if $p \geqslant 2q + r_n$, with $q$ a positive integer, then there is $T \gt 0$, depending only on the norms of the initial and boundary data, such that the Einstein equations (1.1) has a unique (up to a diffeomorphism) solution $g$ on $(-T, T) \times M$ with the above initial and boundary data, which is such that $x^2 g \in C^{n-1} ((-T, T) \times \overline{M}) \; \cap \; C^q_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$. Furthermore, if $x^2 \widetilde{g} , x^2 K$ are polyhomogeneous of class $C^{\infty}$ and $\widehat{g}$ is in $C^{\infty} ((-T_0, T_0) \times \partial \overline{M})$, then $x^2 g$ is in $C^{\infty}_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$.
</p>projecteuclid.org/euclid.jdg/1563242472_20190715220128Mon, 15 Jul 2019 22:01 EDTGenus bounds for min-max minimal surfaceshttps://projecteuclid.org/euclid.jdg/1563242473<strong>Daniel Ketover</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 112, Number 3, 555--590.</p><p><strong>Abstract:</strong><br/>
We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon–Smith. This confirms a conjecture made by Pitts–Rubinstein in 1986.
</p>projecteuclid.org/euclid.jdg/1563242473_20190715220128Mon, 15 Jul 2019 22:01 EDTNonlinear stability results for the modified Mullins–Sekerka and the surface diffusion flowhttps://projecteuclid.org/euclid.jdg/1567216953<strong>E. Acerbi</strong>, <strong>N. Fusco</strong>, <strong>V. Julin</strong>, <strong>M. Morini</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 1, 1--53.</p><p><strong>Abstract:</strong><br/>
It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins–Sekerka or Hele–Shaw flow. The same result holds for three-dimensional periodic configurations that are strictly stable with respect to the sharp-interface Ohta–Kawaski energy. In this case, they are exponentially stable for the so-called modified Mullins–Sekerka flow.
</p>projecteuclid.org/euclid.jdg/1567216953_20190830220250Fri, 30 Aug 2019 22:02 EDTStable blowup for the supercritical Yang–Mills heat flowhttps://projecteuclid.org/euclid.jdg/1567216954<strong>Roland Donninger</strong>, <strong>Birgit Schörkhuber</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 1, 55--94.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the heat flow for Yang–Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)$-equivariant setting, the Yang–Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove [“Singularity formation in the Yang-Mills flow”, Calc. Var. Partial Differential Equations , 19(2):211–220, 2004]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.
</p>projecteuclid.org/euclid.jdg/1567216954_20190830220250Fri, 30 Aug 2019 22:02 EDTMaximizing Steklov eigenvalues on surfaceshttps://projecteuclid.org/euclid.jdg/1567216955<strong>Romain Petrides</strong>. <p><strong>Source: </strong>Journal of Differential Geometry, Volume 113, Number 1, 95--188.</p><p><strong>Abstract:</strong><br/>
We study the Steklov eigenvalue functionals $\sigma_k (\Sigma, g) L_g (\partial \Sigma)$ on smooth surfaces with non-empty boundary. We prove that, under some natural gap assumptions, these functionals do admit maximal metrics which come with an associated minimal surface with free boundary from $\Sigma$ into some Euclidean ball, generalizing previous results by Fraser and Schoen in [“Sharp eigenvalue bounds and minimal surfaces in the ball,” Invent. Math. , 203(3):823–890, 2016].
</p>projecteuclid.org/euclid.jdg/1567216955_20190830220250Fri, 30 Aug 2019 22:02 EDT