All Questions
Tagged with dg.differential-geometry fa.functional-analysis
313 questions
2
votes
0
answers
122
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 forms....
- 51
31
votes
7
answers
4k
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 ...
- 17.5k
9
votes
1
answer
557
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 ...
- 91
3
votes
1
answer
686
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 ...
- 3,612
5
votes
0
answers
584
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 ...
user34072
3
votes
1
answer
328
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 $x_i$-...
- 33
3
votes
1
answer
661
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 ...
- 1,303
3
votes
1
answer
733
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.
...
10
votes
1
answer
957
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 $L$ ...
- 103
1
vote
1
answer
260
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 ...
- 339
1
vote
1
answer
283
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 $\...
- 5,824
6
votes
1
answer
753
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 ...
- 225
7
votes
0
answers
199
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 ...
- 1,040
2
votes
2
answers
326
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 $S$ ...
- 55
13
votes
3
answers
2k
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 1.3.9)...
- 5,824
1
vote
1
answer
154
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 (...
- 1,727
2
votes
1
answer
2k
views
Monge–Ampère operator
I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't
understand the proof of the following proposition.
Let $u$, $v$ be plurisubharmonic functions defined ...
- 29
1
vote
0
answers
346
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} = g(...
- 93
2
votes
2
answers
513
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, ...
- 23
7
votes
1
answer
4k
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.
...
- 515
3
votes
2
answers
1k
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 ...
- 49
1
vote
1
answer
534
views
Unique symplectic form in an adapted complex structure
I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. ...
- 13
4
votes
1
answer
426
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) = \...
- 1,829
6
votes
0
answers
324
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. ...
- 61
1
vote
2
answers
622
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 ...
- 29
2
votes
1
answer
261
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 $C^k(...
- 31
1
vote
2
answers
318
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 $\Omega-B(...
- 25
-3
votes
1
answer
634
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!
- 25
2
votes
1
answer
687
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 = \...
12
votes
0
answers
478
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 ...
- 8,803
3
votes
1
answer
254
views
gluing along a real analytic manifold
hi,
I have a general question. Assume we have a real analytic $n-$dim. manifold $X$ and $M$ a real analytic compact submanifold of $X$ (of dimension less that the dimension of $X$, say $k < n$). ...
- 89
0
votes
0
answers
606
views
partial differential equations with mixed boundary conditions
hi,
does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ?
actually I am intrested in the following: Let $f(x)=(f_{1}(x),...,f_{n}(...
- 89
8
votes
1
answer
1k
views
derivative in the Wasserstein space
Villani gives the following formula to find the gradient of a function $F$ of a probability density function $\rho$ in the Wasserstein space :
$$\nabla_W F(\rho) = -\nabla.(\rho \nabla \frac{\delta F}{...
- 481
7
votes
2
answers
920
views
Exotic spectrum of Laplace operator
Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
- 101
7
votes
1
answer
1k
views
laplace equation on manifolds with boundary
in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f $ if ...
- 213
2
votes
0
answers
524
views
What essential property justifies the name "derivative"?
Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
- 643
7
votes
2
answers
1k
views
Yang Mills gradient/heat flow on 4-torus
The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,
$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta ...
- 362
4
votes
3
answers
2k
views
looking for a book on banach manifolds
Hi,
I am looking for a book on Banach manifolds. Can somebody recommend me something.
Thanks in advance.
leo
- 41
8
votes
0
answers
196
views
Parametrizing derivations from the algebra of smooth functions on a manifold to its dual
$\newcommand{\Der}{\operatorname{Der}}$
$\newcommand{\Real}{{\mathbb R}}$
(Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
- 25.8k
2
votes
2
answers
560
views
Will the eigenvalue of the dirac operater tend to negative infinity?
Question: If M is a spin manifold. Condider the dirac operator on a spinor bundle. Can the eigenvalue of this operater tend to negative infinity? If it can, can we choose a riemann matrix such that ...
- 283
9
votes
1
answer
682
views
A differential inequality needed to prove a theorem about odd-dimensional souls
I need a solution to this problem (which is really a calculus problem) in order to prove a rigidity result for open nonnegatively curved manifolds with odd-dimensional souls:
Suppose that $f,g:\...
- 93
7
votes
4
answers
973
views
I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps ...
- 5,053
8
votes
2
answers
865
views
frechet manifolds book
hi, does anyone know a good book or some lecture notes on the theory of frechet manifolds ?
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^...
- 73
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 ...
user11178
7
votes
1
answer
1k
views
How to construct a scalar differential operator having the same spectrum as a non-scalar differential operator exploiting symmetries?
I am interested in eigenvalue problems for differential operators acting on one forms on closed two-dimensional manifolds and how they relate to eigenvalue problems of associated operators acting on ...
5
votes
2
answers
1k
views
Sobolev imbedding on Riemannian manifolds
Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy.
Let $W^{1,...
- 397
7
votes
1
answer
1k
views
dependence of eigenvalues on parameters
Let $f$ be a positive real-analytic function on the closed unit disk. Consider the eigenvalue problem $\Delta \phi = \lambda f \phi$,
with $\phi = 0$ on the boundary. There exists a sequence of ...
- 1,233
24
votes
3
answers
2k
views
The third axiom in the definition of (infinite-dimensional) vector bundles: why?
Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
- 243
16
votes
3
answers
5k
views
Integration of differential forms using measure theory?
Setup: Let $(M,g)$ be a (possibly non-compact) Riemannian manifold with volume density $d_gV$. Then one may think of $(M,g)$ as a measure space $(\Omega,\mathcal{A},\mu)$, where $\Omega:=M$, $\mathcal{...
- 408