6
votes

Accepted

### Reference for $\epsilon$-regularity

First, it is straightforward to adapt Haslhofer's proof to higher dimensions, using the Sobolev inequality: For any compactly supported $u$ such that $\|\nabla u\|_2 < \infty$,
$$ \|u\|_{\frac{2n}{...

5
votes

Accepted

### Horizontal knots on 3 sphere

The plane field orthogonal to the Hopf fibration is a “standard contact structure” on the three-sphere. Knots whose tangent plane lies in this contact structure are called Legendrian knots. These ...

4
votes

### Smooth isometric immersions of the a hemisphere in $\mathbb R^3$

Actually, it's not enough to specify the isometric immersion along a curve. You have to specify more data than that. For example, you can reflect the unit sphere across its equator, and that will ...

4
votes

Accepted

### Are Bergman metrics on compact Riemann surfaces continuous on Teichmüller space?

Yes, these Riemannian metrics are continuous. In fact, much better is true: they are real-analytic on the Teichmüller universal curve $\mathscr{C}(R)$. This is a direct consequence of Theorem III of ...

3
votes

Accepted

### Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces

Karpelevich and Mostow independently proved that there exists an equivariant totally-geodesic embedding $f: X_1\to X_2$ (irreducibility is irrelevant, all you need is compactness of the kernel). By ...

2
votes

### Is a Riemannian submersion a harmonic map?

I just realize that the answer is negative:
Radu Pantilie, Some remarks on harmonic Riemannian submersion, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série ...

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