# Tag Info

1 vote

### Isoperimetric inequality for minimal surfaces bounded by space curves containing a line segment

The area minimizing surface $\Sigma$ with boundary $\gamma$ has nonpositive curvature in the sense of Alexandrov. Applying Reshetnyak majorization theorem (see 9.56), we get a convex plane figure $F$ ...
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### Coherent sheaves and holomorphic vector bundles

I think the general idea of "coherent sheaves are what you get what you try to make the category of vector bundles into an abelian category" is a good intuition to have, and this is true for ...
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• 101k
Accepted

### Is an inextensible manifold necessarily compact?

Yes, $M$ must be compact. In fact, if $M$ is non-compact, it admits a non-surjective self embedding $f:M\rightarrow M$. When $n=1$, the only non-compact manifold is $\mathbb{R}$, which obviously ...
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### Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

This result is stated as Exercise 7 in Chapter 8 of Bröcker and Jänich's book Introduction to Differential Topology (p. 86 of the English translation). This may or may not count as a citeable ...
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### Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

The result in the following paper implies that open star-shaped domainin $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. But, in your case, a diffeomorphism can be obtained along the same lines. K. ...
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### Comparing diffusion processes in different metrics

The generator of the diffusion corresponding to a Riemannian metric (i.e., the diffusion process which the limit of the random walks such that their increments go along geodesics and the ...
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### A question on a result of Colin de Verdière

Vedrin Šahović in his unpublished thesis [Approximations of Riemannian manifolds with linear curvature constraints, 2009] proved that any compact metric space can be appoximated by hyperbolic ...
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### Topology of the space of embedded genus $g$ surfaces in $S^3$

The first observation is that $\mathcal{E}_g$ is not connected when $g > 0$. This is due to the existence of "knotting". For example, in genus one, let $K$ and $K'$ be smooth knots. ...
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Accepted

### Intersection of conical neighbourhoods on a polyhedral space

You say "The same is true for a tubular neighborhood of the edge". This is not correct, but it is true if you stay away from the endpoints. So $U$ should be defined as a tubular neighborhood ...
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### Minimize the area of a maximal surface in R^{2,1} with boundary on the unit sphere

That's a very interesting question but I don't think that the estimate as it is proposed can hold (as suggested also in the update at the end of the question). (We just discussed this with Nathaniel ...
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### Is it possible to average a riemannian metric over an action and preserve curvature bounds?

As it was mentioned by Igor Belegradek, the curvature gets destroyed by averaging. Assume there is a modified the averaging process that preserves positive curvature. Then you would prove Hopf ...
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1 vote
Accepted

### Continuity of a reaching time of a submanifold

If $V$ is closed in $\mathcal{O}$, then $\tau^V$ is continuous at $x_0$. If $V$ is not closed, then it is not hard to find counterexamples (e.g. imagine $\mathcal{O}=\mathbb{R}^3$, $V$ is an open disk ...
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1 vote

### trapped geodesics

Let us shoot a geodesic from a random point in a random direction. Note that with probability 0 it is one-sides finite + other-side infinite. However, if you shoot a geodesic from a nonregular point, ...
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### A characterization of round sphere

Suppose $\gamma$ is a geodesic in $M$. Then there is a function $\ell$ such that $|\gamma(t_0)-\gamma(t_1)|=\ell(|t_0-t_1|)$, assuming that $|t_0-t_1|$ is small. It follows that $\gamma$ has constant ...
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Accepted

### Smooth mapping from $\mathrm{RP}^2$ to $\mathbb{R}^3$ with nonsingular derivative

This is a famous problem that was solved by a doctoral student of David Hilbert named Werner Boy in 1901. The kind of mapping you are looking for is called an "immersion" (of its domain — ...
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Accepted

### Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy

For torus knots, all of the representations into $SU(2)$ were rather explicitly worked out by Eric Klassen (Representations of knot groups in $SU(2)$. Trans. Amer. Math. Soc. 326 (1991), no. 2, 795–...

### Usefulness of Nash embedding theorem

Let me add that the generalized Gauss–Bonnet formula was first proved for embedded Riemannian manifolds. It was done independently in Weyl's "On the volume of tubes" (1939) and Allendoerfer'...
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Accepted

### Space of spacelike embeddings as infinite-dimensional manifold

A standard reference on infinite dimensional manifolds is Kriegl, Andreas; Michor, Peter W., The convenient setting of global analysis, Mathematical Surveys and Monographs. 53. Providence, RI: ...
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