Related papers: Gradient Estimates on Dirichlet Eigenfunctions

Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $\Delta_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_\lambda$ be $L^2$-normalized eigenfunctions…

For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $\Lambda_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $\Lambda_gf=\left.\frac{\partial…

In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $\lambda_{F}(p,\Omega)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal…

Let $M$ be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets $D$ in $M$ for which the corresponding…

Let $(X,G)$ be a $d$-dimensional compact smooth Riemannian manifold equipped with Laplace-Beltrami operator $\Delta_{G}$, and let $\Pi_{X}$ be the $C^{\ast}$-algebra obtained by locally transferring the $C^{\ast}$-algebra generated by…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

Let $(M, g, f, \tau)$ be a complete Ricci shrinker satisfying $\textrm{Ric}+\nabla^2f=\frac{g}{2\tau}$ and let $R$ denote its scalar curvature. For a confined function $V$ on $M$, we obtain a lower bound for the lowest eigenvalue of the…

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2…

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_\lambda: \Delta_M \varphi_\lambda + \lambda \varphi_\lambda = 0$ denote the Laplace eigenfunction on $\mathbb{M}$…

We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds…

We prove a gradient estimate for Donaldson's equation \[\omega\wedge(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{n-1}=e^F(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^n\] (and its parabolic analog) on an $n$-dimensional compact…

We classify all self-dual Einstein four-manifolds invariant under a principal action of the three-dimensional Heisenberg group with non-degenerate orbits. The metrics are explicit and we find, in particular, that the Einstein constant can…

Let $(M,g)$ and $(M',g')$ be non-orientable Riemannian surfaces with fixed boundary $\Gamma$ and fixed Euler characterictic $m$, and $\Lambda$ and $\Lambda'$ be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of…

We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$. We prove there exists a unique solution using the continuity method which…

We give a new lower bound for the first gap $\lambda_2 - \lambda_1$ of the Dirichlet eigenvalues of the Schr{\"o}dinger operator on a bounded convex domain $\Omega$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new…

We prove a Yau's type gradient estimate for positive $f$-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry-Emery Ricci tensor and the weighted…

We study the principal Dirichlet eigenfunction $\varphi_U$ when the domain $U$ is a perturbation of a bounded inner uniform domain in a strictly local regular Dirichlet space. We prove that if $U$ is suitably contained in between two inner…

In this paper we consider the inverse problem of determining on a compact Riemannian manifold the electric potential and the absorption coefficient in the wave equation with Dirichlet data from measured Neumann boundary observations. This…

In order to minimize a differentiable geodesically convex function, we study a second-order dynamical system on Riemannian manifolds with an asymptotically vanishing damping term of the form $\alpha/t$. For positive values of $\alpha$,…

We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to…