Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$ $$ \int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|} $$ by Gauss-Bonnet and th …

Let $ f $ be in $ L^1(\Omega) $ where $ \Omega $ is an open subset of $ \mathbb{R}^n $. Also, assume that $ B(x_i,r_i) $ is a collection of disjoint open balls in $ \Omega $ such …

First let me state a result of Kazdan and Warner Let $M$ be a compact orientable two dimensional manifold. Let $f:M \rightarrow \mathbb{R}$ be a function that has the same sig …

First let me state a Theorem due to Kazdan and Warner: ``Let M be a compact two dimensional orientable manifold. Let $f: M \rightarrow \mathbb{R}$ be a function that has the sam …

Hello, I've been wondering today around the following exercise in J.Heinonen's "Lectures on Analysis on Metric Spaces" (see below for terminology): prove that the statement of Vit …

I'm sure that many readers are already familiar with the well known Bonnesen inequality in the plane for a smooth, connected curve: $(R_{out} - R_{in})^2 \leq \pi^2 (L^2 - 4\pi A …

A catenary curve is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is: What is the s …

Hi everybody, I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than top …

A friend in physics asked this question, and I didn't know the answer. Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric …

We approach the problem of finding a metric of constant curvature on a surface (i.e. a $C^\infty$ 2-manifold). Specifically, what we want to do is, given a surface $M$ and a metric …

The $\varepsilon$-regularity phenomenon shows up in several different contexts. I try to describe it focussing on the harmonic map situation, but I really would like to understand …

Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then th …

Take an infinitely long rope of diameter 1, ideally flexible and ideally slippery. Tie a trefoil knot into it and pull it tight. Describe the resulting rope shape analytically. Th …

Let $M$ be a closed Riemannian manifold and $\Delta$ its Laplace-Beltrami operator. Then we have Yau's estimate on the first (non-zero) eigenvalue $\lambda_1>0$ of $\Delta$ acting …