# Tagged Questions

210 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
116 views

### Third variation of area of a minimal surface

There is a formula for the third variation of area on page 96 of Nitsche's book, Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the page it is good for normal ...
253 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...
368 views

### Triple bubble conjecture: Natural candidate?

Is there a standard natural candidate surface for the shape that encloses three given volumes in $\mathbb{R}^3$ and has minimal surface area? I know the planar triple bubble conjecture was ...
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### Surfaces with many (but not solely) closed geodesics?

Let $S$ be a closed surface embedded in $\mathbb{R}^3$, let's say of genus zero. I seek examples of $S$ with the following property: If one selects a random any point $p$ on $S$, and a random ...
198 views

### Some facts about cut-locus

Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic to $S^2$. S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree." How the set of point can be a tree? ...
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### Integral of Square of Mean Curvature

Let us assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus. (1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$. (2)When ...
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### Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
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### How to compute the normals to Costa's minimal surface?

I am trying to draw Costa's minimal surface in high resolution using the PovRay raytracer. For this I need to compute points on the surface as well as the normals. It is relatively easy to compute the ...
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### Geodesics on the twisted pseudosphere (Dini's surface)

I wonder how difficult it is to compute geodesics on Dini's Surface, a twisted pseudosphere? Here is one parametrization, from Alfred Gray's Modern Differential Geometry of Curves and Surfaces, ...
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### Costa's minimal surface and the structure of lungs

Seeing this image of Costa's minimal surface        (MathWorld image) made me wonder if the fine-grained structure of the human lung is somehow composed of pieces of ...
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### Fixing a proof of the systolic inequality for higher genus surfaces

I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap in one of the ...
Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator ...