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I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and c …

In the case of a regularly-sampled scalar-valued signal $f$ on the real line, we can construct a discrete linear operator $A$ such that $A(f)$ approximates $\partial^2 f / \partial x^2$. One way to i …

Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct eige …

Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional $$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$ where $X_p(f)$ is the dir …

The Question Let $M$ be a compact, connected, orientable surface without boundary of bounded genus smoothly embedded in $\mathbb{R}^3$; define the diameter $d_M$ of $M$ as the maximum minimal (geodes …

Consider the heat equation $$\dot{u} = \Delta u$$ with initial conditions $$u_0 = \delta(x)$$ for some point $x$ in the domain $\Omega$ of the problem. If $\Omega$ is $\mathbb{R}^n$, then this pr …

Much has been said about bounds on Laplacian eigenvalues, and the literature can be tough to sort through! I am specifically interested in the case where the domain is a closed compact surface, and am …