User paul - MathOverflow

Control of the $C^1$ norm of a diffeomorphism

2013-02-05T09:47:37Z

Let $\Omega$ be a smooth open set of $\mathbb{R}^3$ diffeomorphic to the unit ball $B$. Let assumme that the boundary $\partial \Omega=\Sigma$ is also smooth and satisfies:

$$\int_\Sigma H^2 d\sigma \leq C,$$ where $H$ is the mean curvature of $\Sigma$. Does there exists a diffeomorphism for $\Omega$ to $B$ whose $C^1$ is controlled by a cosntant which depends only on $C$.

This question have been tackle in here in the case of a rotationnaly invariant domain, using the fact that in dimension $2$ we control the the diffeomorphism between a ball and a simply conected domain assuming that the curve which bound the domain satisfy a chord-arc condition.

Is there is any references about this problem?

Detail: Let $\Omega$ as above and one diffeomorphism $\psi$ from $\Omega$ to $B$, do we have $$ \inf_{\phi\in Diffeo(B)} \vert \nabla (\phi\circ \psi)\vert \; \leq K,$$ where $K$ depends only on $C$.

In fact, i am also searching a reference for the problem in dimension $2$ (perhaps here we can replace diffeomorphism by conformal diffeomorphism) replacing $\int_\Sigma H^2 d\sigma \leq C$ by $\int_\Gamma \kappa^2 d\sigma \leq C$ where $\Gamma=\partial \Omega$.

I am ok if "global geometric" hypothesis are needed but only "global geometric"( i.e. about volume, area, length, total curvature).

growth of energy of eigenfunctions on hyperbolic surface

2012-11-21T13:07:36Z

I am looking to the behavior of eigenfunctions associated to small eigenvalues on a degenerating hyperbolic surface.

Let $(\Sigma_n, h_n)$ a sequence of compact surfaces with area equal to $1$ and curvature equal to $-1$. Let $\phi_n$ an eigenfunction associated to $\lambda_n$ which goes to $0$, such that $\Vert \phi_n\Vert_2=1$. Hence we easily prove that $\Vert \nabla \phi_n \Vert_2 =\sqrt{\lambda_n}$. But I guess that the energy in smaler in the thick part, I guess we have $\Vert \nabla {\phi_n}_{\vert K} \Vert_2 =O(\lambda_n)$ on every $K\subset \Sigma_n$ where the injectivity radius in bounded from below.

My idea was to study $\phi_n$ in the collar using the fact this region is isometric to

$\mathbb{H} / {z\mapsto e^l z} $. Then in polar coordinate $$\phi_n(r,\theta)=\sum_n a_n(\theta) r^{\frac{2\pi i n}{l}}$$
where $$a_n" +\left( \frac{\lambda_n}{sin^2(\theta)} -\left(\frac{2\pi n}{l} \right)^2\right)a_n=0$$

whose solutions are given by Legendre functions and then I try to get some estimates on the growth of the energy. Unfortunately I didn't succeed and I didn't find references about this precise subject. Since I am not a specialist of this field, I tryed to read classical references such as Buser book or the paper of Wolpert 'Spectral limit for hyperbolic surfaces' which studies notably the growth of the $L^2$ norm for eigenfunction associated to eigenvalue bigger than $1/4$.

So I am looking for any new ideas or references on that questions. Thanks in advance, Paul

degenerating surface II

2012-01-09T15:34:09Z

In http://mathoverflow.net/questions/81325/degenerating-surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is the following: Can a sequence of constant mean curvature (non-zero a prior) converge (in $C^2$- topology) to a branch immersion if we impose a condition on the curvature of the boundary such that $$\int \kappa d\sigma< 4\pi .$$

Indeed, there is some regularity results for minimal surface which exclude branch point under this kind of assumption, see §377 of the book of Nitsche on minimal surface, moreover here the mean curvature can be made small by rescaling the ambient space since the condition on the the total curvature is scale-invariant. My feeling is that it is impossible to smoothly converge to a branch immersion without a complicated boundary or curvature.... but i have no proof.

degenerating surface

2011-11-19T08:42:03Z

Hi, i have a sequence of immersed disc $u_n: \mathbb{D} \rightarrow \mathbb{R}^3$ which converge to a singular cover of the disc: $z^k$ for $k\geq 2$, moreprecisely $u_n \rightarrow z^k$ in $C^2(\mathbb{D})$. Of course the Gauss curvature of the image $\Sigma_n=u_n(\mathbb{B})$ blows up thanks to Gauss-Bonnet formula : $\int_{\Sigma_n}K= (1-k)2\pi + o(1)$. My questions are the following:

1) can the Gauss curvature be bounded form above? i.e the blow-up come only from necks and there is no pinching region... my feeeling is no, since you have to close the surface.

2) Extra bonus: same question with only a convergence in $C^2_{loc}(\mathbb{D}\setminus { 0})$, here we allow the closing of the surface be made by a big sphere for example.

Answer by Paul for Finding constant curvature metrics on surfaces for the case of positive Euler characteristic

2010-11-17T07:13:59Z

If you relax the fact the metrics $g$ and $g_0$ have to be point-wise conformal to be globally conformal, i.e there exists $\phi$ a diffeomorphism of $S^2$ such that $g=e^u \phi^*(g_0)$. Then the existence of a metric conform to $g_0$ with constant curvature is equivalente to the fact that there is only one conformal class on $S^2$. Which can be proved using the fact every surface is locally conformaly flat and the fact that $S^2$ is simply connected.

Answer by Paul for Curves of constant curvature on S^2

2010-11-17T07:01:32Z

In a recent paper, Sun proved that

i)such curves concentrate around the critical point of the Gaussian curvature

ii) there exits a curve with constant geodesic curvature in every neighborhood of a non-degenerate critical of the Gaussian curvature

My intuition is that we have

ii') there exits a foliation of a neighborhood of non-degenerate critical of the Gaussian curvature foliated by curves with constant geodesic curvature and this foliation is unique

Since we have such a result for surface with constant mean curvature, see Ye91 and Ye96.

So this gives a picture of the asymptotic structure of this moduli space as a one dimensional manifold. However i guess that that the question of the global structure is quite open.

Answer by Paul for Sobolev imbedding failure due to a kink in the domain

2010-09-18T08:55:46Z

Hi, I don't know such an example, but the weaker condition i know is the following

The Weak Cone Condition: Given $x\in \Omega$, let $R(x)$ consist of all points $y\in \Omega$ such that the line segment from $x$ to $y$ lies in $\Omega$. Let $$\Gamma(x)={y \in R ( x )\hbox { s.t. } \vert y - x \vert < 1}.$$ We say that $\Omega$ satisfies the weak cone condition if there exists $\delta > 0$ such that $$ \vert (\Gamma(x))\vert \geq \delta \hbox{ for all } x\in \Omega .$$

You will find a proof in: Adams, R. A.; Fournier, John Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61 (1977), no. 3, 713--734.

Comment by Paul

2013-02-05T16:28:31Z

In general yes, But here we search the best diffeomorphism from $\Omega$ to $B$. Unless you have a counterexample?

Comment by Paul

2012-11-22T13:41:57Z

Thank you for the reference. The proof is clearer than in Wolpert but it "just" prove that $\phi_n$ converge to some $\phi_*$(in my case a constant). I would like to know the rate of convergence on the thick part.

Comment by Paul

2012-01-15T13:56:39Z

i mean the curvature of the curvature of the boundary curve seen as an embedded curve of $\R^3$. In Your example it is something like $2\pi(2k+1)$.

Comment by Paul

2011-11-23T09:57:42Z

For a clear expository text of viscosity solution in the case of elliptic equation you can have look to the chapter 5 of the book of Qing Han and Fanghua Lin, Elliptic Partial Differential Equations, you have many useful estimate as Harnack inequality or $W^{2,p}$-estimate.

Comment by Paul

2011-11-22T22:15:29Z

Thank you Vitali and Robert for your proof of the fact an immersion is possible if and only if $k$ is odd.