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8 votes
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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
4 votes
0 answers
159 views

inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
X-Naut PhD's user avatar
3 votes
0 answers
219 views

Strictly contracting solutions to the Eikonal equation on Riemannian manifolds

Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$. Question: Does there exist, on every complete ...
Nate River's user avatar
  • 6,205
3 votes
0 answers
65 views

Elliptic equations in semi-infinite strips

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
Ali's user avatar
  • 4,145
2 votes
0 answers
120 views

Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator $$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$ There are two ...
B.Hueber's user avatar
  • 1,171
2 votes
0 answers
188 views

Self-adjointness of fractional laplacian

Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
B.Hueber's user avatar
  • 1,171
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
2 votes
0 answers
85 views

Proving an eigenvalue bound without resorting to Weyl's law

Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
Ali's user avatar
  • 4,145
2 votes
0 answers
77 views

Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator

I would appreciate any answers or even references for the following problem. Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
Ali's user avatar
  • 4,145
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
2 votes
0 answers
55 views

Approximate a one-form with nowhere vanishing one-forms having bounded Laplacians

This is a follow-up question of this one. Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a ...
Asaf Shachar's user avatar
  • 6,741
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
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
1 vote
0 answers
157 views

Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...
Asaf Shachar's user avatar
  • 6,741
1 vote
0 answers
100 views

singular integral operators

Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator. My ...
Ali's user avatar
  • 4,145
0 votes
0 answers
60 views

The size of super level sets and the symmetry on a sphere

Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define $$ S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}. $$ Suppose ...
MathLearner's user avatar
0 votes
0 answers
59 views

Gauss lemma for nonsmooth metric

$g_{ij}(x)\in L^\infty(\mathbb{R}^n, M^{n\times n})$ is a metric in $\mathbb{R}^n$ satisfying $\lambda |x|^2\leq g_{ij}x^ix^j\leq \Lambda |x|^2$($\lambda>0$&$\Lambda>0$) Can we find a ...
Yuchen Bi's user avatar
  • 101
0 votes
0 answers
145 views

Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
Lucy's user avatar
  • 183