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This MSE question contains a slick proof of the fact that If $i:M^n\hookrightarrow \mathbb{R}^{n+1}$ is an embedded submanifold, then writing $g=i^*\delta$ as the induced metric on $M$, the volume …

One striking example of the failure of Hamilton-Ivey pinching can be seen here in which it is shown that the FIK shrinkers (which do not have non-negative Ricci curvature, much less non-negative secti …

For $\Omega\subset \mathbb{R}^3$ a region with $|\Omega| = |B_1|$, let $$ C(\Omega) = \int_\Omega\int_\Omega \frac{dxdy}{|x-y|} $$ denote the Coulomb (or gravitational, etc) energy. Poincaré is cred …

Here is a set of of measurable functions with cardinality the continuum whose infimum is not (Borel) measurable: Let $S\subset [0,1]$ be a non-measurable set. For $t\in[0,1]$ let $f_t(x)$ be the func …

Here is an explicit example, which may or may not fit into your requirements: In http://www.ams.org/mathscinet-getitem?mr=308905, "The equivariant Plateau problem and interior regularity," Lawson sho …

There's a paper of Berard Besson Gallot who generalize the Levy--Gromov result to have a diameter dependence as well as allowing for negative lower curvature bounds: "Sur une inégalité isopérimétriqu …

The comments section was getting unwieldy, so I'll answer here. Hopefully this is helpful. What I was trying to say is as follows: suppose that $(M,g,X)$ is a steady gradient soliton, i.e. $$ \math …

Below, I mean smooth oriented closed connected manifolds and smooth maps (but am happy to hear about the topological category, or unoriented manifolds, etc instead). Say that $X^n$ has the Hopf proper …

From the work of Lott--Villani and Sturm, I know that the following fact holds: (*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci curvatur …

The answer is yes (I'm assuming you are asking about closed manifolds, non-compactness allows for all sort of crazy things to happen, you can check out the work of Topping and collaborators). Kotsch …

EDIT: As pointed out by Tapio Rajala, my proof is wrong without assuming that $X$ is compact. I've added this assumption, which I don't think is necessary, but I am having some trouble seeing how to d …

Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I …

Yang and Yau proved that for a surface of genus $\gamma$, $\Sigma$ with a metric $g$, the first eigenvalue satisfies $$ \lambda_1(g) Area(g) \leq 8\pi (1+\gamma). $$ So, the answer to your first ques …

There's been several articles in the comments that are "historical survey" articles. Its not totally clear if you're interested in "current research surveys," but if you are, here are several very ni …

An interesting "curvature" which has recently received much interest is the "Ma--Trudinger--Wang" (MTW) tensor, which arose in the study of when optimal transport maps are smooth on a Riemannian manif …