微分几何

We use the generalized Gauss-Bonnet formula for Riemannian polyhedra discovered by Allendoerfer, Weil and Chern to show that hyperbolic space of dimension $n$ has no isometric immersion into Euclidean space of dimension $2n-1$.

We show that a compact oriented riemannian four-manifold with harmonic and pinched self-dual Weyl curvature is anti-self-dual if the type is nonpositive. The main part is to show that there is an almost-K\"ahler structure outside the zero…

In this paper, after explaining some basic aspects of the modern theory of foliations with the aim of describing the celebrated Reeb foliation, we propose the first construction of a comprehensive physical model of it. The construction of…

In this paper, we study systematically the concentration properties of Finsler metric measure manifolds. We establish the relationships between the concentration properties and the observable diameter, isoperimetric inequalities and the…

This paper resolves a long-standing open problem by providing a classification of Willmore $2$-spheres in $S^n$. We show that any such $2$-sphere is either totally isotropic--originating from the projection of a special twistor curve in the…

The purpose of this note is describe and classify the splittable lattices in the completely solvable metabelian Lie group (semidirect product of abelian vector groups) $G:=\mathbb{R}^n\rtimes_\eta\mathbb{R}^m$, where $\eta$ is the…

We show that a Lie group $G$ admitting a bi-invariant distance must be the product $G=H\times K$ of an abelian group $H$ and a compact group $K$ with discrete center. Moreover, the distance in $G$ must come from the infima of lengths of…

In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, any biharmonic function of subquadratic growth must be…

Diffeology extends differential geometry to spaces beyond smooth manifolds. This paper explores diffeology's key features and illustrates its utility with examples including singular and quotient spaces, and applications in symplectic…

We continue the~study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a~paper by Gol$'$dshtein and Troyanov. As in this paper, our study is based on relations between $L_{q,p}$-cohomology…

We consider compact oriented four-manifolds with harmonic self-dual Weyl curvature in addition to a pinching condition.

For any Hermitian holomorphic vector bundle with Nakano positive curvature tensor, Demailly introduced a Monge-Amp\`ere type equation. When the rank of the bundle is $1$, it becomes the usual Monge-Amp\`ere equation. In this paper, we solve…

We study unknottedness for free boundary minimal surfaces in a three-dimensional Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary, and for self-shrinkers in the three-dimensional Euclidean space. For doing…

The Willmore Problem seeks closed surfaces in $\mathbb{S}^3\subset\mathbb{R}^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |H_{\mathbb{R}^4}|^2 = area + \int |H_{\mathbb{S}^3}|^2$. The longstanding…

In this paper, we introduce a natural notion of constant curvature Lorentzian surfaces with conical singularities, and provide a large class of examples of such structures. We moreover initiate the study of their global rigidity, by proving…

We prove various comparison theorems of the $i$-th eigenvalue $\lambda_i$ of the Laplacian on fibred Riemannian manifolds by using fiberwise spherical and Euclidean (or hyperbolic) symmetrization. In particular we generalize the…

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact…

Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a…

In this paper, we prove a Li-Yau-Hamilton type Harnack estimate for the $f$-mean curvature flow in Euclidean space, which can be viewed as a gradient flow of the weighed area functional with the measure density function $e^{-f}$.

We consider Riemannian manifolds $(M^n,g_0)$, $(M^n,h)$, where $(M^n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0 < \infty$, and $(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$ for some small…