1
vote
0answers
124 views

Cotangent bundle of symmetric space is symmetric space?

Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...
3
votes
1answer
119 views

Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds

Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an ...
11
votes
3answers
657 views

Sobolev spaces and geometry

This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study ...
0
votes
0answers
89 views

Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)

Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
2
votes
1answer
92 views

Practical way to check whether a distribution is conormal

Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that $$ L_1 ...
8
votes
0answers
228 views

Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...
1
vote
2answers
132 views

How to show this integral on boundary of Lipschitz domain is finite?

Sorry for asking a basic question but this did not get answered on M.SE. Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that $$\int_{\partial\Omega} ...
1
vote
1answer
131 views

If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lipschitz hypersurface)?

Let $M$ be a bounded hypersurface. Let $f \in H^{\frac 12}(M)$ and let $\varphi\colon M \to \mathbb{R}$ be a Lipschitz function. When $M=\Omega \subset \mathbb{R}^n$ an open domain, we know that the ...
3
votes
0answers
85 views

construction of heat kernels for non-compact manifolds with boundary

Recently, I am studying heat semigroup for noncompact manifolds with boundary. In Issac Chavel's book "eigenvalues in Riemannian geometry". "Given a noncompact Riemannian manifold, it need not be ...
4
votes
1answer
462 views

A question on Grassmannian

Let $V$ be the space of $4$ by $4$ Hermitian matrices, that is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform measure of $$ \left\{ W\in Gr\left(5,V\right):W \text{contains no ...
5
votes
3answers
227 views

When does heat kernel have both Gaussian upper and lower bounds?

Recently, I am reading Sturm's paper "Analysis on local Dirichlet form III: X is a locally compact separable Hausdorff space and m is a positive Radon measure with supp[m]=X. $\varepsilon$ is a ...
1
vote
1answer
124 views

heat kernel $p_t(x_0,y) \in D(\Delta) \cap L^\infty$ for a manifold with Ricci curvature bounded below?

X is an n-dim Riemannian manifold with the Dirichlet form $$ \varepsilon (u,v) =-\int_X \langle \nabla u,\nabla v \rangle $$ for $u,v \in W^{1,2}(X)$. Let $P_t$ and $p_t(x,y)$ be the associate ...
-3
votes
2answers
355 views

Are there examples of compact infinite dimensional manifolds? [closed]

Are there known examples of compact infinite dimensional manifolds? The word "manifold" is important.
0
votes
0answers
110 views

When is $X^*d$ skew-symmetric?

here $X$ is a vector field on a manifold $M$ and $d$ is the usual deRham operator. The motivation of the question comes from a differential geometry homework problem which asked to prove something if ...
4
votes
2answers
264 views

Can we define smooth diffeomorphisms on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature?

Can we define smooth diffeomorphisms on the Hilbert cube$[0,1]^\mathbb{N}$ ? Has it been done in the literature? For example, let $E$ be the set of bijections $f$ of the Hilbert cube, such that ...
10
votes
1answer
336 views

Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric. It is essentially ...
4
votes
1answer
262 views

Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support. Is the reverse true? Namely that if some PDO ...
2
votes
1answer
255 views

Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...
2
votes
0answers
84 views

A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential ...
21
votes
7answers
838 views

Intuition for failure of Implicit Function theorem on Frechet Manifolds

When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds ...
8
votes
1answer
200 views

Dimension of eigenspaces of Laplacian on a compact Riemannian manifold

Let $M$ be a compact smooth manifold, let $g$ a riemannian metric and let $\Delta_{g}$ the Laplacian operator on functions induced by $g$. Is there a (topological?) bound on the dimension of $n$-th ...
3
votes
1answer
171 views

Delta-convex functions and inner products

A delta-convex (d.c.) function is one which can be written as the difference of two convex functions. The space of d.c. functions includes all C2 functions, and is interesting because it allows many ...
0
votes
0answers
102 views

Sobolev spaces on hypersurfaces

I am learning about Sobolev spaces on hypersurfaces. Let $S$ be a $C^k$-hypersurface with boundary for some $k$. In order to define a weak derivative, one needs $k \geq 2$ because the integration by ...
3
votes
0answers
174 views

Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...
2
votes
1answer
248 views

Laplacian on coset spaces

Edited after @J. Martel's comment: Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$). We know that if $X_i$ represent the vector fields on $S^2$ giving the rotation about the ...
3
votes
1answer
210 views

What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
2
votes
1answer
224 views

Trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold. ...
0
votes
0answers
202 views

Functional Analysis and Differential Manifold incompatibility

From the mathematical point of view, where is the idea of the incompatibility between functional analysis, at the basis of quantum mechanics, and differential geometry, at the basis of general ...
1
vote
0answers
212 views

Question about the deformation of de Rham-Hodge operator

Let $M$ be an n-dimensional closed oriented Riemannian manifold. For any $e\in\Gamma(TM)$, let $e^{*}\in\Gamma(T^{\ast}M)$ corresponds to e via $g^{TM}$. Let $\hat{c}(e)$ be the Clifford operator ...
10
votes
1answer
436 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...
0
votes
1answer
149 views

Flat connection, finite-dimensional space of covariant constant one forms

hallo, I have the following question: Let $U\subset \mathbb{R}^{n}$ be an open subset. Furthermore, let $\nabla$ be a flat connection on $U$ (not necessary Levi-Civita). How can one show that the ...
1
vote
1answer
125 views

Cotangent bundle in the category of locally convex spaces

I'm trying to understand the definition of a differential form on $M$ in the context of Fréchet spaces or, more generally, locally convex spaces. The standard procedure defines a k-form as a map ...
4
votes
1answer
277 views

Banach Manifold

Let $M$ and $N$ be closed manifolds. Is it true that $C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
6
votes
0answers
116 views

Central Extension of Continuous Loop Group

For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
1
vote
2answers
223 views

Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold?

Hi, I am looking for the result: $$\text{The norm} \quad \lVert \Delta u \rVert_{L^2(S)} \quad \text{is equivalent to} \quad \lVert u \rVert_{H^2(S)}$$ for scalar functions $u \in H^2(S)$, where $S$ ...
8
votes
3answers
467 views

Space of sections of a fibre bundle with non-compact base space

Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$. For compact $M$ it is well known (Hamilton 1982, Part II Corollary ...
1
vote
1answer
97 views

Question about coercivity of a functional

Hi! Let $(M,g)$ be a compact Riemannian manifold without boundary of dimension $2m$. Let $$T:W^{2,2}(M)\rightarrow L^{2}(T^{*}M\otimes TM)$$ be a second order, linear, differential operator ...
1
vote
0answers
167 views

HyperKaehler manifolds are Ricci-flat

Hi, I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = ...
2
votes
2answers
306 views

Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold

Hallo, I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, ...
3
votes
1answer
596 views

Functional/variational derivative and the Leibniz rule

I am currently trying to understand the BV-formalism, which makes heavy use of the functional derivative. Let us consider the functional derivative, as defined in for example its Wikipedia article. ...
3
votes
2answers
603 views

Infinite dimensional manifold

In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as ...
0
votes
1answer
205 views

Unique symplectic form in an adapted complex structure

Hallo, I ave the following question: Due to Stenzel, Lempert, Szöke ect. we know that a Riemannian manifold $(M,g)$ admits a complex structure on an neighbourhood of the cotangent bundle. This ...
3
votes
1answer
242 views

Smooth functions tangent to the leaves of a foliation

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space $$T_f C^\infty(M,N) = ...
4
votes
0answers
185 views

Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings, Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...
1
vote
1answer
201 views

Kähler manifold with Ricci-flat Kähler form

hallo, I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
2
votes
1answer
213 views

Is C^{k+1}(X) compactly contained in C^{k}(X) for a closed manifold X?

Hi all, I apologize if this question is too low level for mathoverflow. I'm happy to move it to math.stackexchange if so. Let $X$ be a closed manifold, let $k$ be a nonnegative integer and let ...
1
vote
2answers
174 views

Poisson modification of subharmonic function

Let $u\in C^2(\Omega)$ be such that $\Delta u \ge 0$ on $\Omega\supset \overline{B(a,r)}$. We consider the Poisson modification $U$ of $u$ for the ball $B(a,r),$ that is $U$ equals $u$ on ...
-3
votes
1answer
253 views

compactly supported harmonic functions [closed]

Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist? Thanks!
1
vote
1answer
445 views

Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...
7
votes
0answers
198 views

What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...