All Questions
72 questions
3
votes
0
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108
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A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
CommunityBot
- 1
2
votes
0
answers
92
views
Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$
\begin{...
- 356
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)...
- 35.5k
2
votes
0
answers
114
views
Poincare inequality on the hemisphere
Background:
Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
- 537
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 ...
- 131
5
votes
0
answers
360
views
Injectivity of div–curl operator
$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system
\begin{align*}
Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\
u &= 0 \...
- 419
1
vote
0
answers
135
views
Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...
- 969
13
votes
3
answers
2k
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 PDE'...
- 4,713
3
votes
1
answer
214
views
Convergence of spectrum
Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$.
Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
- 45k
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 ...
- 383
7
votes
0
answers
80
views
Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator
Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
- 808
3
votes
0
answers
110
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On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections
Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
4
votes
0
answers
199
views
Spectral problems with the wrong sign on the Poincaré disk
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
- 63.9k
3
votes
0
answers
130
views
Is the range of the exterior covariant derivative closed in $L^{2}$?
Let $(M,g)$ be a compact Riemannian manifold. Given a tensor bundle $\mathbb{E}$, let $\nabla:\Gamma(\mathbb{E}) \rightarrow \Gamma(T^{*}M\otimes \mathbb{E})$ be the canonical connection induced by ...
- 808
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 ...
- 1,713
17
votes
1
answer
1k
views
Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)
Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates
\...
CommunityBot
- 1
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 $\...
- 99
1
vote
0
answers
127
views
Laplacian on the sphere and Moving Plane method
Consider the sphere $\mathbb{S}^n$ as a subset of $\mathbb{R}^{n+1}$, thus $\mathbb{S}^n=\{\omega\in \mathbb{R}^{n+1},\sum_{i=1}^{n+1}\omega_i^2=1\}.$
I am interested in studying positive solutions to ...
- 537
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 ...
- 6,367
0
votes
1
answer
147
views
The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]
Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
- 135
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 ...
- 1,942
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 ...
- 24.2k
2
votes
0
answers
382
views
Poincaré inequality holds on Riemannian manifolds (min max principle)
In YuChang Xia's book "Eigenvalues on Riemannian Manifolds" Page 4 equation (1.16）Poincaré inequality:
I want to know which manifold(s)/function(s) can make the inequality hold. What if we ...
- 63.9k
3
votes
0
answers
125
views
Green operator of elliptic differential operator and radius of convergence
Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
- 1,383
2
votes
1
answer
652
views
Extension of outer unit normal vector to interior
Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...
- 28k
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
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
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
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
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
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
$$\partial_t \...
- 277
8
votes
1
answer
311
views
Laplacian spectrum asymptotics in neck stretching
Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
- 3,212
1
vote
0
answers
184
views
One question about Schrodinger Semigroups-(B. Simon)
This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
- 1,915
3
votes
2
answers
956
views
Hodge decomposition on open manifold
For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold.
Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
- 5,984
2
votes
1
answer
352
views
Poisson equation on noncompact manifold
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space.
For the equation
$$\Delta u=f,$$
...
- 41
8
votes
1
answer
712
views
Pseudo-differential operators with compactly supported symbols
If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO $...
- 509
3
votes
2
answers
262
views
The gradient $\nabla u$ of $u\in W^{1,p}(M;N)$ is tangent to $N$ almost everywhere
Let $M,N$ be (compact) Riemannian manifolds. $N$ is viewed as an embedded submanifold of $\Bbb R^K$. The Sobolev space $W^{1,p}(M;N)$ is defined as
$$
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
1
vote
0
answers
304
views
Harmonic coordinates on asymptotically flat manifold
I am studying the existence of harmonic coordinates at infinity on an asymptotically flat manifold. My Reference papers are, The Mass of Asymptotically Flat Manifold, by Bartnik [B] and The Yamabe ...
- 914
1
vote
1
answer
488
views
Motivation behind the parabolic metric
I've been reading some papers about parabolic evolution problems between manifolds. We want to study the behaviour of maps from the domain $(0,T)\times \Omega$, where $\Omega\subset \Bbb R^m$, to some ...
- 5,407
1
vote
1
answer
728
views
Elliptic regularity of Laplace-Beltrami operator on a manifold
I am currently trying to prove an elliptic regularity type result for the Laplace Beltrami operator $\Delta_g$ on a Riemannian manifold $(M^n,g)$. As a matter of convention, I will assume $\Delta_g$ ...
- 156
3
votes
0
answers
280
views
Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
- 1,385
2
votes
1
answer
219
views
Right inverse of the Seiberg-Witten functional
For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e.
$$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...
- 17.5k
0
votes
0
answers
119
views
Gauge Fixing Problem on Cylindrical
For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
- 1,915
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 ...
- 41
4
votes
1
answer
497
views
Eigenfunction basis of Laplacian on a manifold
It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
- 329
5
votes
1
answer
233
views
Zeta-Determinant Theorem
Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...
CommunityBot
- 1
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 $\Delta$...
- 601