For a complete noncompact Riemannian manifold with smoothly bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to such noncompact manifolds.

We give a short proof of Penner-Grushevsky-Schumacher-Trapani’s large genus asymptotics of Weil-Petersson volumes of moduli spaces of curves. We also study asymptotic expansions for certain integrals of pure $\psi$ classes and answer a question of Mirzakhani on the asymptotic behavior of one-point volume polynomials of moduli spaces of curves.

We showed the existence of non-radial solutions of the equation $\Delta u - \lambda u + \lambda u^q = 0$ on the round sphere $S^m$, for $q \lt (m + 2)/ (m - 2)$, and study the number of such solutions in terms of $\lambda$. We show that for any isoparametric hypersurface $M \subset S^m$ there are solutions such that $M$ is a regular level set (and the number of such solutions increases with $\lambda$). We also show similar results for isoparametric hypersurfaces in general Riemannian manifolds. These solutions give multiplicity results for metrics of constant scalar curvature on conformal classes of Riemannian products.

On Kähler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov’s relative volume comparison, Bonnet-Meyers theorem, and Yau’s gradient estimate for positive harmonic functions. The tool is a Bochner type formula reflecting the Kähler structure.

Let $X_0$ be a smooth hypersurface (not assumed generic) in projective space $\mathrm{P}^n$, $n \geq 3$ over the complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in the deformations of the pair $C_0 , X_0$. In this paper, we prove that if the first order deformations of the pair exist along certain first order deformations of the hypersurface $X_0$, then the twisted normal bundle $N_{C_0/ X_0}(1) = N_{C_0 / X_0} \otimes \mathcal{O}_{\mathcal{P}^n} (1) \vert {}_{C_0}$ is generated by global sections.

In our previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exists at least one minimal isoparametric hypersurface. In this paper, we show such a minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian manifold are minimal. Further, we study isoparametric hypersurfaces with constant principal curvatures in general Riemannian manifolds. It turns out that in this case the focal submanifolds have the same properties as those in the standard sphere, i.e., the shape operator with respect to any normal direction has common constant principal curvatures. Some necessary conditions involving Ricci curvature and scalar curvature are also derived.

We give a new example of a curve $C$ algebraically, but not rationally, uniformized by radicals. This means that $C$ has no map onto $\mathbb{P}^1$ with solvable Galois group, while there exists a curve $C'$ that maps onto $C$ and has a finite morphism to $\mathbb{P}^1$ with solvable Galois group. We construct such a curve $C$ of genus $9$ in the second symmetric product of a general curve of genus $2$. It is also an example of a genus $9$ curve that does not satisfy condition $S(4, 2, 9)$ of Abramovich and Harris.

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k : k^p] = p^e \lt \infty$ and $V$ be a $p$-adic representation of $\mathrm{Gal}(\overline{K} / K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K} / K)$ if and only if $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K^\mathrm{pf}} / K^\mathrm{pf})$ where $K^\mathrm{pf} / K$ is a certain $p$-adic local field whose residue field is the smallest perfect field $k^\mathrm{pf}$ containing $k$. As an application, we prove the $p$-adic monodromy theorem of Fontaine in the imperfect residue field case.

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.