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Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as $$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad …

Given $(M^n, g)$ closed riemannian manifold, I am wondering about the definition of the $C^k$ norms with respect to the metric, and how these norms depend on $g$. For example, I would assume that if $ …

I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$, $$\text{diam}(\Sig …

Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by $$y^2 …