All Questions
Tagged with dg.differential-geometry elliptic-pde
152 questions
4
votes
1
answer
145
views
The order of the solution of Liouville equations at singularity
If I consider a Liouville equations in the unit disk $D \setminus\{0\} \subset \mathbb{R}^2$ with singularity at $x=0$,
$$\Delta u= e^{2u}$$
If I define the order of $u$ at origin is defined to be
$$...
- 54
6
votes
2
answers
1k
views
The adjoint operators as elliptic operators
Edit:
It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally ...
- 356
3
votes
1
answer
154
views
Foliation by Asymptotic lines
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. I am interested about existence of surfaces $\Gamma$ embedded in $M$ with the following property:
$\Gamma$ is foliated by geodesics (...
- 4,135
3
votes
0
answers
278
views
Principal eigenvalue of Laplacian under volume preserving mean curvature flow
Consider a compact uniformly convex n-dimensional hypersurface $M_0$ without boundary , which is smoothly imbedded in $\mathbb R^{n+1}$ , and suppose that $M_0$ is represented locally by some ...
- 165
3
votes
0
answers
147
views
Prove the positivity of the subelliptic operator heat kernel
Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition at every point $x\in\mathbb{R}^{n}$, Let $W$ be a ...
- 287
9
votes
1
answer
539
views
Most general maximum principle for non-integrable almost complex structures
Let $H\subseteq\mathbb C^n$ be a smooth co-oriented real codimension one hypersurface. If $H$ is weakly pseudo-convex, then holomorphic maps $u:\Delta\to\mathbb C^n$ ($\Delta$ denotes the unit disk) ...
- 18.7k
7
votes
2
answers
400
views
Is there an upper bound on dimension of kernel of elliptic operator for a fixed closed manifold M
Assume that $M$ is a smooth closed manifold and $E,F$ are fixed smooth vector bundles over $M.$
Is there a number $C,$ such that for any elliptic operator $\mathcal{D}:\Gamma(E)\to\Gamma(F)$
$$\...
- 1,615
2
votes
1
answer
477
views
Any Good Reference for Kazdan-Warner Type Equations
The Kazdan-Warner Type equations I am talking about is the following:
Suppose X be a compact Riemannian manifold(of any dimension) and $A$, $B$, and $w$ to be smooth function, I hope to know more ...
- 703
4
votes
0
answers
259
views
Does a "symbolically elliptic" sequence of operators have an analytic index?
Does a symbolically elliptic sequence of differential operators have an analytic index? cohomology? For example, is there any concrete meaning of the Todd genus of an almost complex manifold in terms ...
- 8,717
3
votes
1
answer
605
views
how to use the sobolev inequality to obtain the embedding theorem
I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. In this article, the authors proved the following theorem
(Theorem 2.3) Let ...
- 287
2
votes
1
answer
292
views
The relationship about sub-unit ball and sub-elliptic ball
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n}$ with smooth boundary.$\{X_{1},\cdots,X_{m}\}$ be smooth real vector fields on $\Omega$ Which satisfy the Hormander condition. If $\gamma$ is ...
- 287
2
votes
0
answers
166
views
Paneitz-Branson operator and Q-curvature
Let $(M,g)$ a n-dimensional Riemannian manifold, $n\neq 4$, and $h=u^{\frac{4}{n-4}}g$. Then the formula that connect the Panitz-Branson opertator $P_g$ and the Q-curvature $Q_{h}$ is
$Q_h=\frac{2}{...
- 61
1
vote
0
answers
113
views
upper bound for heat kernel of Grushin operator
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n+1}$ with smooth boundary. $\Omega\cap\{(0,\cdots,0,y)\in\mathbb{R}^{n+1}| y\in \mathbb{R}\}\neq \varnothing$
Consider the sub-elliptic operator
...
- 287
7
votes
1
answer
558
views
minimal surfaces in $S^n$
Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...
- 914
2
votes
0
answers
218
views
Weyl's law for minimal surfaces
I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...
- 1,616
8
votes
1
answer
533
views
Properties of connection Laplacian on vector fields
Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ ...
- 183
3
votes
2
answers
339
views
Converse to Lichnerowicz Vanishing Theorem?
The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...
- 787
8
votes
0
answers
291
views
Deformation of the covariant Laplacian
Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$).
Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
- 5,824
2
votes
1
answer
338
views
Existence of non-constant solutions for this equations
This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
- 601
2
votes
0
answers
156
views
The minimum value of a energy integral
Let $D \subset {\mathbb{R}^3}$ a simple connected open domain with volume $\int_{\bar D} {dV = 1} $. $\varphi :{\mathbb{R}^3} \to \mathbb{R}$ is ${C^1}$, $\varphi (\infty ) = 0
$ and
$${\nabla ^2}\...
- 273
6
votes
2
answers
1k
views
Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds
For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details.
Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen (http://...
- 3,020
2
votes
2
answers
368
views
Double-layer potentials on Riemannian manifolds
Let $M$ be a compact Riemannian manifold, and let $S \subset M$ be a smooth hypersurface which divides $M$ into two domains $D_1$, $D_2$. Let also $g \colon S \to \mathbb R$ be a smooth function (...
- 241
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 ...
- 1,407
12
votes
2
answers
1k
views
Regularity of Hodge Laplacian on bounded domains
I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates $\lVert \omega \rVert_{W^{s+2,p}} \leq c\lVert f\rVert_{W^{s,p}}$, $s\geq 0$ ...
- 317
3
votes
0
answers
134
views
Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$
Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for $...
- 601
2
votes
1
answer
548
views
Does this PDE only have the trivial solution?
Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
- 376
39
votes
5
answers
5k
views
Explicit eigenvalues of the Laplacian
Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...
- 1,101
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 $\Delta$...
- 43
2
votes
1
answer
420
views
Second order estimates of Monge-Ampere equations
In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...
- 21.8k
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 $P = -\Delta + a$, ...
- 11
2
votes
2
answers
442
views
Principal bundles and Subriemannian Geometry
In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution $\...
- 9,853
3
votes
1
answer
453
views
On fundamental solutions to Poisson equation on 3-dimensional manifolds
I am interesting in solutions to Poisson equation
$$\triangle \varphi = 4 \pi \rho \qquad (1)$$
defined on 3-dimensional oriented Riemannian manifold $(M,g)$,
where $g$ is metric and $\...
- 371
1
vote
1
answer
195
views
Uniqueness affine curvature
Let $\gamma_1,\gamma_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their (special)affine curvature, $\mu_1,\mu_2$ are equal, that is $\mu_1(\theta)=\mu_2(\theta)$, for ...
- 15
2
votes
0
answers
200
views
The level set of convolution
Suppose $u\in C^2(\bar \Omega)\cap C^\infty(\Omega)$ is a function which is compact supported on $\Omega$ with $u\equiv 0$ at $\partial\Omega$, and $\Omega\subset \mathbb R^3$.
We also assume that $\...
- 679
9
votes
1
answer
833
views
Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$
A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a ...
- 601
17
votes
4
answers
3k
views
Green's operator of elliptic differential operator
Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
- 376
2
votes
1
answer
202
views
Eigenfunction on surface with boundary
Suppose we have a two-dimensional surface $M$ with smooth boundary $\partial M$. Equip $M$ with a metric $g$ such that the Gauss curvature $K$ of $M$ and geodesic curvature $\kappa$ of $\partial M$ ...
- 633
0
votes
1
answer
631
views
Green's function and eigenvalues with multiplicity
Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
- 587
11
votes
1
answer
692
views
What is the first eigenvalue of $p$-Laplacian on unit sphere $S^n$?
We know that the first eigenvalue of Laplacian on the Riemannian unit sphere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Laplacian on $S^n$?
The $p$-Laplacian ...
- 199
3
votes
1
answer
318
views
Implicit function theorem for operator
I am reading the paper of Convergence of the Yamabe flow for arbitrary initial energy
I am stuck by one part of the paper. Suppose $u_\infty>0$ is a smooth function on $(M, g_0)$ and
$$L_0=\frac{4(...
- 633
0
votes
0
answers
158
views
positive eigenfunction on complete Riemannian manifold
Let $(M^n,g)$ be a complete(non-compact) Riemannian manifold. Consider the positive solution to the equation
$$
\Delta u = u
$$
where $\Delta=\nabla_i \nabla_i$ is negative semi-definite. Is there ...
- 98
6
votes
0
answers
434
views
Laplacians associated to symplectic cohomologies
I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
- 601
2
votes
0
answers
202
views
Bunimovich stadium bouncing ball
http://terrytao.wordpress.com/2008/07/07/hassells-proof-of-scarring-for-the-bunimovich-stadium/
I cannot relate how the eigenfunctions normalizaed correspond to the probabality distribution of the ...
- 163
7
votes
1
answer
2k
views
Yau's conjecture for positive Chern class
I heard in a conference that Yau's conjecture is open for positive Chern class. I read in an article that talked about some stability conditions necessary in this case. So I want to know if this ...
- 2,106
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
27
votes
1
answer
2k
views
Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood?
Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
- 25.3k
6
votes
2
answers
1k
views
Schauder estimates for higher order linear elliptic operator on manifold
Hi!
Let $(M,g)$ be a smooth compact riemannian manifold without boundary. Let $L$ be a linear elliptic operator on $M$ of order $2k$ with smooth coefficients. Suppose i have $u\in W^{2k,2}(M)$ and $f\...
- 1,727
9
votes
0
answers
2k
views
elliptic regularity on manifolds
Hello!
I'm reading the book of T. Aubin Some nonlinear problems in Riemannian geometry. In chapter 3 he introduces elliptic operators on manifolds, but then he gives regularity results for elliptic ...
- 91
4
votes
1
answer
569
views
Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?
Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...
- 3,059
3
votes
1
answer
551
views
Harmonic/conformal map composition between manifolds in either order?
Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is (...
- 705