Related papers: Llarull's theorem on punctured sphere with $L^\inf…

On the two-sphere $\Sigma$, we consider the problem of minimising among suitable immersions $f \,\colon \Sigma \rightarrow \mathbb{R}^3$ the weighted $L^\infty$ norm of the mean curvature $H$, with weighting given by a prescribed ambient…

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…

We give an interpretation of the hemisphere rigidity theorem of Hang-Wang in the framework of Gelfand problem. More precisely, Hang-Wang showed that for a metric $g$ conformal to the standard metric $g_0$ on $S^{n}_{+}$ with $R\geq n(n-1)$…

We determine the curvature equations of natural metrics on tangent bundles and radius r tangent sphere bundles S_rM of a Riemannian manifold M. A family of positive scalar curvature metrics on S_rM is found, for any M with bounded sectional…

A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main…

Suppose that $\theta_1,\theta_2,\dots,\theta_n$ are positive numbers and $n\ge 3$. Does there exist a sphere with a spherical metric with $n$ conical singularities of angles $2\pi\theta_1,2\pi\theta_2,\dots,2\pi\theta_n$? A sufficient…

The Ricci version of the Schur theorem is shown to hold for a wide class of Finsler metrics. What is more, let $F$ be any (positive definite) Finsler metric such that $\text{Ric} =\rho F^2$ with $\rho\colon M^n\rightarrow\mathbb{R}$ (i.e.,…

We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is…

In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros…

Let $\Omega$ be a domain on the unit $n$-sphere $ \mathbb S^n$ and $\mathring{g}$ the standard metric of $\mathbb S^n$, $n\ge 3$. We show that there exists a conformal metric $g$ with vanishing scalar curvature $R(g)=0$ such that $(\Omega,…

We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation…

Here, an extension of the Obata-Tanno's theorem to Finsler geometry is established and the following rigidity result is obtained; Every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to…

We determine the maximal number of systoles among all spheres with $n$ punctures endowed with a complete Riemannian metric of finite area.

We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm…

We first give a precise statement on the short time existence of the Calabi flow and prove a stability result: any metric near a constant scalar curvature metric will flow to this cscK metric exponentially fast. Secondly, we prove that a…

In a previous paper we showed that static spherically symmetric objects which, in the vicinity of their surface, are well-described by a polytropic equation of state with 3/2<Gamma<2 exhibit a curvature singularity in Palatini f(R) gravity.…

We prove for $n\in\{3,4,5\}$ that the connected sum of a closed aspherical $n$-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

Alexandrov's soap bubble theorem asserts that spheres are the only connected closed embedded hypersurfaces in the Euclidean space with constant mean curvature. The theorem can be extended to space forms and it holds for more general…

In this paper, we prove several rigidity results for complete noncompact manifolds with nonnegative intermediate curvatures. We show that when either $3\leq n\leq 5$, $1\leq m\leq n-1$, or $6\leq n\leq 7$, $m\in \{1,n-1,n-2\}$, any manifold…