All Questions
Tagged with dg.differential-geometry fa.functional-analysis
313 questions
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{...
- 35.5k
2
votes
2
answers
447
views
Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$
Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
- 12.3k
3
votes
0
answers
58
views
Criteria for density of subgroup of diffeomorphism group
Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
- 63.9k
18
votes
3
answers
2k
views
Poincare lemma for non-smooth differentiable forms
The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...
- 16.4k
2
votes
1
answer
301
views
Density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
- 5,405
2
votes
0
answers
58
views
Is keeping the kernel fixed an open condition for maps of vector bundles?
More precisely, let $M$ be a smooth manifold, $E_1$, $E_2$ vector bundles over $M$, and consider a $C^\infty(M)$-linear map $A:\Gamma(E_1) \to \Gamma(E_2)$ of vector bundles.
Now consider the ...
- 73
7
votes
0
answers
237
views
Understanding the odd-dimensional index
Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
- 52.3k
5
votes
0
answers
191
views
Index of the Fredholm operator
I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
- 63.9k
13
votes
2
answers
655
views
Random matrix with given singular values
Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let
$$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
- 4,361
2
votes
2
answers
200
views
MP critical point has morse index 1, proof
I wonder where can I find a proof of the following fact: if the mountain pass critical point is non-degenerate, then its Morse index is 1. I am very interested in reading it.
In general, I am ...
- 407
0
votes
0
answers
97
views
Smooth sections of finite dimensional bundle and covering space
Let $G$ be a discrete finitely generated group which acts properly and freely on a smooth manifold $M$ with compact quotient $M/G$. Is it right to consider any function $f \in C^{\infty}_c(M)$ (with ...
- 63.9k
8
votes
1
answer
496
views
Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?
Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space:
$$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
- 3,574
2
votes
1
answer
843
views
Proof of Helmholtz-Hodge decomposition, poor man's version
Helmholtz (-Hodge) decomposition commonly used in physics includes decomposition of a (sufficiently smooth) vector field $F = -\mathrm{grad}(U) + \mathrm{curl}(W)$ on bounded simply connected domain $\...
- 39k
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 ...
- 657
1
vote
1
answer
151
views
A marginal space splitting $\{ \psi \}^{\perp}$
Let $\psi \in L^2(\mathbb R^2,\mathbb C)$. Is there a continuous projection from $\{ \psi \}^{\perp}$ onto
$$
\left\{ \varphi \in L^2(\mathbb R^2) \:\:\Big| \int \overline{\psi}(x,y) \varphi(x,y)\...
- 6,367
3
votes
0
answers
91
views
Pseudodifferential operator associated to a self-adjoint extension of a symmetric operator on an incomplete manifold
Let $D$ be the Dirac operator acting on a spinor bundle $S$ over a complete Riemannian manifold $M$. Then $D$ is an essentially self-adjoint operator on $L^2(S)$.
Suppose there is a compact subset $K\...
- 1,903
4
votes
1
answer
727
views
Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds
On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
- 421
0
votes
0
answers
149
views
Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold
Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
- 3,486
5
votes
1
answer
743
views
Eigenvalues and Domain of the Laplace-Beltrami Operator
Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
- 3,486
2
votes
0
answers
100
views
Regularity for Laplacian operator on non-compact manifold
Let $(M,g)$ be a complete non-compact Riemannian manifold .
Thanks to @EveryLT, we know that the Poisson equation
$$\Delta u=f,$$
is solvable
for some $f\in L^2_k(M)$.
Q Suppose that $(M,g)$ is ...
- 1,915
4
votes
1
answer
229
views
Orientability of moduli space and determinant bundle of ASD operator
Setting
In instanton gauge theory, given a $G$-principal bundle $P\to X^4$, the orientability of the moduli space of ASD connections
$$\mathcal{M}_k = \{A \in L^{2}_{k}(X, \Lambda^1 \otimes\mathrm{...
- 2,533
1
vote
0
answers
123
views
Is this integral zero?
I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation.
Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...
- 327
2
votes
0
answers
551
views
Euler-Lagrange equations on a differentiable manifold
I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
- 651
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 \...
- 277
3
votes
0
answers
53
views
Controlling a Schwartz kernel near the diagonal
Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
- 1,903
2
votes
0
answers
119
views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
- 21
8
votes
1
answer
311
views
Laplacian spectrum asymptotics in neck stretching
Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
- 3,212
1
vote
0
answers
184
views
One question about Schrodinger Semigroups-(B. Simon)
This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
- 1,915
2
votes
1
answer
210
views
$L^{2}$ Betti number
Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...
- 3,486
3
votes
2
answers
956
views
Hodge decomposition on open manifold
For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold.
Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
- 5,984
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/...
- 1,915
2
votes
1
answer
352
views
Poisson equation on noncompact manifold
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space.
For the equation
$$\Delta u=f,$$
...
- 41
5
votes
1
answer
710
views
Smooth structure on the space of sections of a fiber bundle and gauge group
Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...
- 2,533
3
votes
1
answer
243
views
Prescribing a gradient direction
Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...
- 1,675
17
votes
3
answers
770
views
Does a spectral gap lift to covering spaces?
Let $M$ be a complete Riemannian manifold. Denote $\Delta_M\ge0$ the unique self-adjoint extension of the Laplace-Beltrami operator in $L^2(M)$ and $\sigma(\Delta_M)\subset [0,\infty)$ its spectrum. ...
- 331
13
votes
1
answer
465
views
One question about the $\eta$ invariant
This question is from the paper, The Analysis of Elliptic Families
II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8.
Suppose ...
- 1,915
2
votes
0
answers
333
views
The determinant curvature
Let $(M,g)$ be a riemannian manifold and $R(X,Y)$ the riemannian curvature as a two form with values in the endomorphisms of the tangent bundle. I define:
$$
D_g(X,Y)=det(R)(X,Y)
$$
with $det$ the ...
- 187
7
votes
2
answers
460
views
Gaussian Surface Area of Positive Semidefinite Cone
Let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e.g., one that has smooth boundary or is convex. We define the $\epsilon$-neighbor of $A$ in the ...
- 13.6k
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 ...
- 2,789
4
votes
0
answers
120
views
Representation on square integrable sections of a principal bundle
Let $X\rightarrow Y$ be a smooth principal $G$-bundle for some Lie group $G$. Then $L^2(X)$ has a natural $G$-action determined by fibrewise action of $G$ on $X$.
We have an abstract isomorphism of ...
- 1,903
5
votes
2
answers
350
views
Reference Request: Finite dimensional submanifolds of the space of smooth mappings
I apologize for my ignorance, but hope that someone would provide some pointers to what I am sure is a reasonably well-developed body of theory. Consider $C^\infty(U,V)$ where $U \subset R^k$ and $V \...
- 1,675
8
votes
2
answers
1k
views
Example for an integral, rectifiable varifold with unbounded first variation
I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.
Recapitulation
for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
8
votes
1
answer
712
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 $...
- 509
3
votes
0
answers
180
views
When is a minimal immersion holomorphic?
Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let:
$\phi\colon X\to Y$
be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
- 2,816
3
votes
2
answers
262
views
The gradient $\nabla u$ of $u\in W^{1,p}(M;N)$ is tangent to $N$ almost everywhere
Let $M,N$ be (compact) Riemannian manifolds. $N$ is viewed as an embedded submanifold of $\Bbb R^K$. The Sobolev space $W^{1,p}(M;N)$ is defined as
$$
W^{1,p}(M;N):=\{ u\in W^{1,p}(M;\Bbb R^K)\ |\ u(x)...
- 186
4
votes
1
answer
398
views
Proving the inequality $|\nabla |\nabla^r \psi|| \le |\nabla^{r+1} \psi|$
Following Aubin's book "Some nonlinear problems in Riemannian geometry", we use the notation
$$
|\nabla^r \psi|^2 = \nabla_{\alpha_1}\cdots \nabla_{\alpha_r}\psi \nabla^{\alpha_1}\cdots \nabla^{\...
- 856
1
vote
1
answer
633
views
Existence of solution to heat equation on a compact manifold
Let $M$ be a compact Riemannian manifold (without boundary), I would like to know under which regularity conditions can we solve the heat equation
$$\begin{align}
\partial_tu-\Delta u &= f \\
u(\...
- 26.3k
0
votes
1
answer
175
views
Accessible reference for (scattering) $\Psi DO$'s on manifolds
I am currently trying to understand Hassell, Tao, and Wunsch's paper on Strichartz estimates on non-trapping asymptotically conic manifolds, however, my understanding of pseudodifferential operators ...
- 6,249
3
votes
0
answers
108
views
Radial Poincare inequality for Gaussian measures
Let $\mu$ be a zero mean Gaussian probability measure on $\mathbb{R}^n$ whose covariance is less than the identity. If $f$ is a $1$-Lipschitz real function on $\mathbb{R}^n$ such that there exists a ...
- 2,772
7
votes
1
answer
281
views
Harmonic functions on $(M,g)$ closed, induce an embedding in Euclidean space
In Hajime Urakawa's monograph The Spectral Geometry of the Laplacian on page 41, we make an assumption that I can't quite justify on my own. The following is our setup:
Let $(M^n,g)$ be a closed ...
- 4,131