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8 votes
0 answers
103 views

Sobolev embedding theorems in vector bundles on non-compact manifolds

Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
G. Blaickner's user avatar
  • 1,429
6 votes
1 answer
196 views

On elliptic operators on non-compact manifolds

Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
G. Blaickner's user avatar
  • 1,429
4 votes
1 answer
128 views

Lower bound of mean curvature implies that the set is subset of a given ball

If a simply connected set $\Omega\subset\mathbb{R}^n$ has $C^2$ boundary such that the mean curvature $H$ of $\partial \Omega$ satisfies: $$H\geq 1$$ Does this imply that $\Omega\subset B_1$ after ...
Holden Lyu's user avatar
1 vote
0 answers
54 views

Isoperimetric Inequalities in Annular Regions

Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that $$ \left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
MathLearner's user avatar
7 votes
2 answers
626 views

Elliptic regularity on manifolds: Is this true?

Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
B.Hueber's user avatar
  • 1,171
1 vote
1 answer
160 views

Differentiability of an integral of geodesic distance

Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$. Q1: Define $$ g(t)=\...
Hengchao Chen's user avatar
4 votes
1 answer
202 views

Finding a real-analytic diffeomorphism

Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
Ali's user avatar
  • 4,135
5 votes
3 answers
620 views

Poisson equation on manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation $$\Delta u=f$$ does have a solution on $C^{\infty}(\mathcal{M})$ ...
B.Hueber's user avatar
  • 1,171
3 votes
1 answer
340 views

Shrinking a disk with fixed differential

Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential \begin{align} \mathsf{d} f= \begin{pmatrix} \cos\psi(x) &\cos\phi(y) \\ \sin \psi(x)& \sin\phi(y) \end{...
Daniel Castro's user avatar
2 votes
0 answers
148 views

Finding an asymptotically flat manifold with ${\rm Ric}_{r\phi} = \frac{\sin\theta}{r^2}$

Let $(r,\theta,\phi)$ be the spherical coordinates on $\mathbb{R}^3$ where $\theta \in (0,\pi)$ and $\phi\in (0,2\pi)$. Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1$...
Laithy's user avatar
  • 969
1 vote
1 answer
137 views

Smoothness of the asymptotic parametrization of a ruled surface

Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form $$\begin{align} \sigma \colon I \...
Matteo Raffaelli's user avatar
0 votes
1 answer
711 views

Lipschitz domains ambiguous definitions

I use a lot in the study of pde bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^N$. However I have noticed that there are some major differences in their definitions. I will put here two of them, ...
Bogdan's user avatar
  • 1,759
1 vote
0 answers
259 views

Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics

Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$? One ...
Laithy's user avatar
  • 969
5 votes
1 answer
339 views

Finding vector fields on $S^2$ with equal divergence

Let $\mathfrak{X}_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}_{CK}$ be the 6-dimensional space of conformal ...
Laithy's user avatar
  • 969
2 votes
1 answer
741 views

Continuity of the perimeter of level sets w.r.t. level function

Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
Bogdan's user avatar
  • 1,759
2 votes
0 answers
109 views

Finding $\Omega$ such that the 1-form $\Omega^2 \omega$ is $L^2$ orthogonal to conformal killing vector fields on $S^2$

Consider the space $\mathcal{A}$ of functions $\Omega$ such that $\Omega^2 \gamma_0$ is isometric to the round sphere, where $\gamma_0$ is the round sphere. (so $\Omega^2 \gamma_0$ is of constant ...
Laithy's user avatar
  • 969
2 votes
0 answers
141 views

For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc

Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying $$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$ where $g_0$, $...
Laithy's user avatar
  • 969
2 votes
0 answers
88 views

$1$-parameter analytic functions are almost everywhere Morse

Let $I = [t_{0}, t_{1}]$ be a closed interval with $t_{0} < t_{1}$ and let $M$ be a compact real analytic $n$-dimensional manifold without boundary. Furthermore, let $f:I \times M \rightarrow \...
Bene's user avatar
  • 21
2 votes
0 answers
85 views

Are a map with constant singular values and its inverse always conjugate through isometries?

Let $U \subseteq \mathbb R^2$ be open, connected and bounded, and let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1 \sigma_2=1$. Suppose that $f:U \to U$ is a diffeomorphism whose singular values (of $...
Asaf Shachar's user avatar
  • 6,741
8 votes
2 answers
452 views

Vector field with constant divergence around embedded submanifold

Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$. Question: Does there ...
StanleyT's user avatar
4 votes
1 answer
224 views

When is the cut-locus normal coordinate collared

Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold, $p \in M$ be fixed and let $C_p$ be the cut-locus of $p$. Other than when $M$ is non-positively curved (in which $C_p= \emptyset$ by ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
320 views

Riemannian submanifolds of Euclidean space admitting Lipschitz extension of Lipschitz functions, and converse statement

Let $M \subset \mathbb{R}^p$ be a Riemannian submanifold. In what follows, when we talk about a Lipschitz function $f$ on $M$, namely $f: M \to \mathbb{R}$, we will assume there is a $L > 0$ so ...
Learning math's user avatar
14 votes
2 answers
871 views

Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine?

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map whose differential has fixed distinct singular values $0<\sigma_1<\sigma_2$ and an everywhere positive determinant (which is the product $\...
Asaf Shachar's user avatar
  • 6,741
4 votes
3 answers
3k views

Covariant derivative of determinant of the metric tensor

Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...
Phillip's user avatar
  • 131