For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all nonflat two-dimensional self-shrinkers. This confirms a conjecture of Colding, Ilmanen, Minicozzi, and White in dimension two.

It has long been suggested that solutions to the linear scalar wave equation

$${\square }_g\varphi =0$$ on a fixed subextremal Reissner–Nordström spacetime with nonvanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W_{\mathrm{loc}}^{1,2}$. This instability is related to the celebrated blue-shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein–Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner–Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price’s law decay is generically sharp along the event horizon.

A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure (in the sense of local differential geometry) in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure—the pairwise nonintegrability condition and the bracket-generating condition—we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X\subset {\mathbb{P}}^N$ is a Fano submanifold of Picard number $1$ and the family of lines covering $X$ becomes a web. In this special case, we have the stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X,X\text{'}\subset {\mathbb{P}}^N$, $dimX\text{'}\ge 3$, are two such Fano manifolds of Picard number $1$, then any surjective morphism $f:X\to X\text{'}$ is an isomorphism.

We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties—reminiscent of the inclusion-exclusion principle for the cardinal on finite sets—of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the exponential generating sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. A direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an Euler–Maclaurin formula. The compatibility with subdivisions of the convolution quotient arising in the algebraic Birkhoff factorization then yields the Euler–Maclaurin formula for any lattice cone. This provides a simple formula for the interpolating factor by means of a projection formula.

We prove that in a simple real Lie group, there is no Borel measurable dense subgroup of intermediate Hausdorff dimension.