Related papers: Some sharp estimates for convex hypersurfaces of p…

In this paper, we investigate Hessian curvature hypersurfaces with prescribed Gauss images. Given geodesically strictly convex bounded domains $\Omega$ in $\mathbb{R}^n$ and $\tilde{\Omega}$ in the unit hemisphere, we prove that there is a…

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to…

Given a pseudoconvex domain U with C^1-boundary in P^n, n>2, we show that if H^{2n-2}_\dR}(U)\not=0, then there is a strictly psh function in a neighborhood of boundary U. We also solve the \dbar-equation in X=P^n\ U, for data smooth (0,1)…

We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume…

We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $\Sigma$ is rotationally symmetric in the following…

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In…

In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply…

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 and conjectured similar results holds for spaces with…

We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to…

We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume - and therefore on the volume of the domain - must fail. Here we…

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is…

The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be…

In this paper, we present a proof of Schauder estimate on Euclidean space and use it to generalize Donaldson's Schauder estimate on space with conical singularities in the following two directions. The first is that we allow the total cone…

In our prior work toward Bartnik's static vacuum extension conjecture for near Euclidean boundary data, we establish a sufficient condition, called static regular, and confirm large classes of boundary hypersurfaces are static regular. In…

We derive some estimates for stable minimal hypersurfaces in $R^{n+1}$. The estimates are related to recent proofs of Bernstein theorems for complete stable minimal hypersurfaces in $R^{n+1}$ for $3\le n\le 5$ by Chodosh-Li,…

Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth…

Exploiting the special features of four-dimensional Riemannian geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5. First, we provide a complete description of the Weyl tensor for…

We show that all non-developable ruled surfaces endowed with Ricci metrics in the three-dimensional Euclidean space may be constructed using curves of constant torsion and its binormal. This allows us to give characterizations of the…

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…