# Tagged Questions

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### An isoperimetric type maximization problem with a barrier.

I'm trying to minmize a particular functional which depends on a curve with fixed endpoints which lies below a fixed line in $\mathbb{R}^2$. Here are the details: Let $(r(\theta), \theta)$ be a ...
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### What are the most general types of curves in $\mathbb{R}^2$ for which Gauss-Bonnet holds?

I would like to know what is the most general form of the Gauss-Bonnet theorem in the plane for curves. It is well known for that for any piecewise $C^2$ simply connected curve with corners, one has ...
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### Is there a characterization of generalized constant mean curvature surfaces?

It is a well known result of Alexandrov that the only compact, connected, constant mean curvature surface is the ball. There is a generalized notion of curvature known as generalized mean curvature ...
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### Minimal surface which divides a convex body into two regions of equal volume

Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume? Background/motivation. A 2D version of ...
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### Generalization of First Variation of Area

The area of an $m$-rectifiable varifold in $n$-dimensional space can be expressed in terms of the surface divergence. More precisely, if $M$ is $m$-rectifiable, $\Omega$ is open, $\eta$ is a $C^1_c$ ...
Let $\Gamma$ be the set of all closed $C^2$ curves in the plane which enclose unit area and let $\Omega$ be the set of all subsets of $\mathbb{R}^2$ that are enclosed by some curve in $\Gamma$. Now ...