相关论文: On a manifold formulation of self-concordant funct…

We prove that in a Riemannian manifold $M$, each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere $S^n$ and the hyperbolic space…

In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $\lambda_2 - \lambda_1$ of Dirichlet eigenvalues, amoung domains with fixed $\lambda_1$. We prove an upper bound on $\lambda_2 - \lambda_1$ for…

We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we…

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the…

We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also…

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain…

The contracting boundary of a proper geodesic metric space generalizes the Gromov boundary of a hyperbolic space. It consists of contracting geodesics up to bounded Hausdorff distances. Another generalization of the Gromov boundary is the…

The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove…

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a…

The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral…

If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\dist^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance…

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the…

A rather complete investigation of anisotropic Bessel potential, Besov, and H\"older spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends'…

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles. The geodesic cycles we study are themselves hyperbolic manifolds of lower dimension. The restriction of an automorphic form to…

We study the geodesic convexity of various energy and entropy functionals restricted to (non-geodesically convex) submanifolds of Wasserstein spaces with their induced geometry. We prove a variety of convexity results by means of a simple…

This paper presents necessary, sufficient, and equivalent conditions for the spherical convexity of non-homogeneous quadratic functions. In addition to motivating this study and identifying useful criteria for determining whether such…

In this paper, we prove that convex hypersurfaces under the flow by powers $\alpha>0$ of the Gauss curvature in space forms $\mathbb{N}^{n+1}(\kappa)$ of constant sectional curvature $\kappa$ $(\kappa=\pm 1)$ contract to a point in finite…

We bound the $L^2$-norm of an $L^2$ harmonic $1$-form in an orientable cusped hyperbolic $3$-manifold $M$ by its topological complexity, measured by the Thurston norm, up to a constant depending on $M$. It generalizes two inequalities of…

Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…