Related papers: On eigenfunction restriction estimates and $L^4$-b…

In this paper we provide a way of taking $L^p$, $p > \frac{m}{2}$ bounds on a $m-$ dimensional Riemannian metric and transforming that into H\"{o}lder bounds for the corresponding distance function. One can think of this new estimate as a…

Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $\lambda_{1,p}$ on $M$, namely $$ e^{f}div(e^{-f}|\nabla v|^{p-2}\nabla…

In this paper, we provide the lower bounds of the first non-zero basic eigenvalue on a closed singular Riemannian manifold $(M,\mathcal{F})$ with basic mean curvature that depends on the given non-negative lower bound of the Ricci curvature…

Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and…

In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schr\"odinger equations) on a compact, smooth Riemannian manifold,…

In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or…

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$. We will assume that the isotropy $H$-module $\mathfrak {g/h}$ has a simple spectrum, i.e. irreducible submodules are…

In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an $n$-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the…

In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with…

Let $M$ be a smooth, compact, connected, oriented Riemannian manifold, and let $\imath: M \to \mathbb R^d$ be an isometric embedding. We show that a Sobolev map $f: M \to M$ which has the property that the differential $df(q)$ is close to…

We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete…

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ \phi_\lambda $ on a closed Riemannian manifold $ (M,g) $. First, in the real-analytic case we present an improvement of…

Geodesic convexity (g-convexity) is a natural generalization of convexity to Riemannian manifolds. However, g-convexity lacks many desirable properties satisfied by Euclidean convexity. For instance, the natural notions of half-spaces and…

We prove a uniform extension result for contracting maps defined on subsets of Hadamard manifolds subject to curvature bounds.

Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature…

Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $\psi$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $\lambda$. We give a new proof to obtain a power saving for the global…

Let $f$ be a Petersson normalized Hecke-Maass cusp form with spectral parameter $t\geq 2$ and let $\mathcal{C}_{D}$ be the union of closed geodesics in $\text{Sl}_{2}(\mathbb{Z})\setminus \mathbb{H}$ associated to a fundamental discriminant…

On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(\lambda\_k(\tau,\alpha))\_k$ of the magnetic Laplacian with the Robin boundary conditions. Here, $\tau$ is a positive number that defines the…

We develop a general deformation principle for families of Riemannian metrics on smooth manifolds with possibly non-compact boundary, preserving lower scalar curvature bounds. The principle is used in order to strengthen boundary…

Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the…