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16 votes
2 answers
1k views

Examples of Banach manifolds with function spaces as tangent spaces

I have recently been learning the theory of Banach manifolds through Serge Lang's book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the ...
proba_124's user avatar
  • 161
15 votes
2 answers
888 views

Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?

Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms $$ \Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
David Roberts's user avatar
  • 35.5k
12 votes
1 answer
2k views

Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...
Benjamin's user avatar
  • 2,099
12 votes
1 answer
727 views

Schemes over topological rings

I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ...
Jonathan Gleason's user avatar
9 votes
1 answer
2k views

Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
Guest's user avatar
  • 123
8 votes
2 answers
1k views

Survey papers on the role played by PDE in mathematics

There are already several questions on MathOverflow that inquire about the many diverse relationships between PDE and several other 'areas' of mathematics (e.g., algebraic and differential geometry ...
8 votes
1 answer
421 views

$C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
SMS's user avatar
  • 1,407
7 votes
2 answers
1k views

A book on Banach Manifold for a Dynamicist

Hi all, Could you give me a suggestion of suitable book about Banach Manifolds for someone that have background in functional analysis at the level of Conway's book and Do Carmo's book on Riemannian ...
user avatar
7 votes
1 answer
1k views

Helmholtz-Decomposition on compact Riemannian manifolds

For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that $L^...
Sören's user avatar
  • 73
7 votes
0 answers
116 views

Bundles over Function Spaces

Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
elena's user avatar
  • 71
6 votes
0 answers
493 views

Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following: Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
uro's user avatar
  • 71
5 votes
3 answers
1k views

Constant rank theorem for Banach spaces

Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
Benjamin's user avatar
  • 2,099
5 votes
1 answer
224 views

Spectral theory of infinite volume hyperbolic manifolds

I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
SMS's user avatar
  • 1,407
5 votes
1 answer
743 views

Eigenvalues and Domain of the Laplace-Beltrami Operator

Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
MartinG's user avatar
  • 51
5 votes
1 answer
328 views

Is a space with p-norm a Finsler manifold?

Suppose $\mathbb{R}^n$ is equipped with the p-norm $\left\Vert x \right\Vert_p$. Let $x\in \mathbb{R}^n$ and let $y$ be in a neighborhood of $x$. The distance between $x$ and $y$ can be defined as $\...
Klock's user avatar
  • 51
4 votes
2 answers
505 views

Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
guest's user avatar
  • 43
4 votes
1 answer
728 views

Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
user144878's user avatar
4 votes
3 answers
669 views

Gaussian bounds on Dirichlet heat kernel

Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0 $ such that $$\frac{c}{t^{n/2}} e^{-\frac{1}{4t}d(x, y)^2} \leq ...
user94178's user avatar
4 votes
2 answers
281 views

Heat kernel asymptotics for small distances

I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies $$p_t(x, y) = \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\...
user85970's user avatar
4 votes
1 answer
532 views

Diffusion semigroup generated by Laplacian

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
anonymos's user avatar
4 votes
0 answers
343 views

Diffeomorphism group action on the space of embeddings

Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...
Hang's user avatar
  • 2,789
4 votes
0 answers
172 views

Donnelly-Fefferman growth of eigenfunctions

Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...
user91752's user avatar
4 votes
0 answers
241 views

"Partition" of a smooth function in $\mathbb R^2$

This is a question asking for reference. I have a proof of the following. Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions $f_1=...
PepeToro's user avatar
  • 231
3 votes
1 answer
255 views

Norm on space of metrics

I recently heard a differential geometry talk where the speaker constructed a one-parameter family of metrics $g(t)$ on a smooth manifold and said that $g(t)$ is real analytic in the Banach space $BC(...
student's user avatar
  • 51
3 votes
1 answer
203 views

Cohomology of the complex of differential forms with Schwartz coefficients

Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
Grisha Taroyan's user avatar
3 votes
1 answer
190 views

Laplace eigenfunction on a polygonal domain symmetric about an axis

Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
user170399's user avatar
3 votes
1 answer
1k views

Friedrichs mollifiers and Sobolev spaces

$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
Carlos Esparza's user avatar
3 votes
2 answers
782 views

Relation between optimal transport cost and difference between topological invariants?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...
Morino_Hikari's user avatar
3 votes
0 answers
153 views

Quasimode construction on a compact Riemannian manifold

Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
Y. Paka's user avatar
  • 131
3 votes
0 answers
479 views

de Rham currents/distributions on manifolds with boundaries

My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
Bence Racskó's user avatar
3 votes
0 answers
153 views

Gaussian heat kernel bounds on Riemannian manifolds [duplicate]

I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$ t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}}, $$ on a ...
user84944's user avatar
3 votes
0 answers
406 views

Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...
user82891's user avatar
2 votes
1 answer
262 views

A clarification regarding analytic perturbation of metrics and Laplacian

This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...
Guest's user avatar
  • 123
2 votes
1 answer
333 views

Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
SMS's user avatar
  • 1,407
2 votes
2 answers
332 views

Heat kernel upper bounds on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds: $$ p(t, x, y) \leq Ct^{-n/2}, \forall x, y, ...
user91989's user avatar
2 votes
0 answers
126 views

Differential equations: trying to connect a nonlinear equation to a linear one

The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
John McManus's user avatar
2 votes
0 answers
255 views

Sobolev Multiplication on non-compact manifold

We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
DLIN's user avatar
  • 1,915
2 votes
0 answers
127 views

Functional inequality under mean curvature flow

Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
Math101's user avatar
  • 143
2 votes
0 answers
234 views

Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\{\varphi_k\}_k \in C^\infty(M)$ such that $\{\varphi_k\}_k$ satisfy the basic concentration ...
SMS's user avatar
  • 1,407
2 votes
0 answers
108 views

Quantitative estimate of heat dispersion - off diagonal estimates

Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = u(...
anonymous's user avatar
1 vote
1 answer
732 views

Norm equivalent to Sobolev norm? [closed]

On the hyperbolic space $\mathbb{H}^n$, it is known that the spectrum of the Laplacian satisfies $\text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty)$. Consider the operator $P = -\Delta + a$, ...
rook's user avatar
  • 11
1 vote
0 answers
56 views

Smooth approximation in Sobolev spaces for surfaces with boundary

Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
user158773's user avatar
1 vote
0 answers
75 views

Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \...
Kei's user avatar
  • 277
0 votes
0 answers
70 views

Normal vector to a level set and fractional Laplacian

Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
user173196's user avatar