All Questions
Tagged with riemannian-geometry elliptic-pde
101 questions
2
votes
0
answers
123
views
Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds
Consider the elliptic operator $Lu = - \Delta u + \langle \nabla u , X \rangle + c \, u $ acting on functions on a closed Riemannian manifold $M$. Here $\Delta$ denotes the Laplace-Beltrami operator, $...
- 386
2
votes
0
answers
113
views
Is this $1$-form harmonic?
Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \...
- 2,095
6
votes
0
answers
91
views
Convergence of free boundary minimal surfaces
I suspect the following statement is true:
Let $(M^3,g)$ be a compact and orientable Riemannian $3$-manifold with nonempty boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and ...
- 2,095
2
votes
0
answers
77
views
Derivative estimates for Laplace eigenfunctions on Riemannian manifolds
In $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$), if $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies $\Delta f+ f=0$, then we can write the following regularity/derivative estimates for $f$:...
- 399
2
votes
0
answers
55
views
Approximate a one-form with nowhere vanishing one-forms having bounded Laplacians
This is a follow-up question of this one.
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a ...
- 6,741
2
votes
0
answers
183
views
Trace operators on submanifolds
In the following paper, Sobolev Spaces on Riemannian Manifolds with Bounded Geometry: General Coordinates and Traces
https://arxiv.org/abs/1301.2539
The authors prove trace theorems for general ...
- 409
5
votes
0
answers
114
views
On boundary-value problems for differential forms on a manifold
Let $M$ be a simply-connected $d$-dimensional Riemannian manifold with boundary (for simplicity assume a ball). Consider the boundary value problem for $\omega\in\Omega^k(M)$,
$$
d\omega = \alpha
\...
- 797
3
votes
1
answer
171
views
Spectrum of the Laplacian on the quotient of $3$-sphere
Given a finite subgroup $\Gamma$ of $O(4)$ acting freely on $S^3$, is there any reference for the spectrum of Laplacian for the transverse-traceless symmetric $2$-tensor on $S^3/\Gamma$ equipped with ...
- 2,535
6
votes
1
answer
577
views
Compactness theorem for minimal surfaces
I am a bit confused about the statement of Theorem 1.1 in this paper by Brian White. For convenience, I will restate it here.
Theorem: Let $\Omega$ be an open subset of a Riemannian $3$-manifold. ...
- 823
0
votes
2
answers
289
views
Derivations of $\chi^{\infty}(M)$ which are elliptic operator
What is an example of a manifold $M$ with $\dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ of smooth vector fields admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such ...
- 356
3
votes
1
answer
162
views
$\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on the boundary and at infinity
Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta_g f = 0$ ...
- 969
2
votes
0
answers
147
views
Calabi $C^3$ estimate
I have a question regarding a computation analogous to the Calabi $C^3$ estimate which is used in the proof of the Calabi--Yau theorem.
Motivation: Establishing Liouville type theorems for complex ...
- 1,379
1
vote
0
answers
62
views
A particular semi-linear equation on Riemannian manifolds
Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation
$$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
- 4,135
5
votes
1
answer
365
views
Regularity of harmonic forms on manifolds-with-boundaries
Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...
- 103
10
votes
1
answer
293
views
Can a harmonic function on a topological cylinder have critical points?
Let $M$ be an oriented closed smooth manifold, and let $C=M\times[0,1]$, the cylinder over $M$. Let $g$ be an arbitrary Riemannian metric on $C$ (in particular, $g$ may look nothing like a product ...
- 203
5
votes
0
answers
307
views
Gradient estimate for Poisson equation on manifold
In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...
- 2,789
3
votes
0
answers
68
views
Diffusion generators with gradient vector fields
Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as
$$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$
where $X_0,X_1,...,X_k$ are ...
- 1,675
11
votes
4
answers
2k
views
Elliptic regularity on compact manifold without boundary
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...
- 571
3
votes
2
answers
373
views
Exponential decay of resolvent kernel
For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
- 9,853
3
votes
1
answer
171
views
Solutions to $\Delta u\ge u^2$
Let $(M,g)$ be a complete Riemannian manifold. Suppose that $u$ is a nonnegative solution to $\Delta_gu\ge u^2$. Does it follow that $u$ must be identically 0?
I know that the answer to above ...
- 93
5
votes
1
answer
162
views
Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?
In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
- 6,741
1
vote
1
answer
293
views
Elliptic regularity of harmonic forms in $L^1$
$\newcommand{\M}{M}$
This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
...
- 6,741
3
votes
1
answer
177
views
Does the space of harmonic forms change continuously with the metric?
Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{...
- 6,741
4
votes
3
answers
522
views
Can we specify the value of harmonic forms at a point?
Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed.
Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$.
Does there exist an open ...
- 6,741
9
votes
0
answers
150
views
Counter-examples to the higher dimensional statement of the half-space theorem
The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$.
The higher dimensional ...
- 823
3
votes
0
answers
105
views
metric with curvature bounded in $L^2$
My question is about the regularity of a metric whose curvature is bounded in $L^2$. Of course, this question doesn't really make sense since the regularity of the metric depends on the coordinates ...
- 914
4
votes
0
answers
136
views
Davies' definition of elliptic operators in "Heat Kernels and Spectral Theory"
I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...
- 5,407
5
votes
1
answer
132
views
Gaussian bounds for the heat kernel of regular domains in Riemannian manifolds
In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same ...
- 5,407
2
votes
1
answer
296
views
Properties of harmonic maps into spheres
Let $(M, g)$ be a complete, noncompact Riemannian $n$-dimensional manifold and let
$\phi \colon M \to \mathbb S^n$ be an harmonic map, where $\mathbb S^n$ is the euclidean $n$-dimensional sphere.
...
- 823
3
votes
1
answer
314
views
Is this approach for establishing regularity of harmonic maps between manifolds valid?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly ...
- 6,741
1
vote
0
answers
142
views
Existence and uniqueness of the Robin problem on a compact, smooth Riemannian manifold with boundary
Let $(\overline{M},g)$ a compact, smooth, Riemannian manifold with boundary $\partial M \in C^\infty$. By $\nu_g$ we denote the normalvectorfield, by $\nabla_g$ the gradient and by $\Delta_g$ the ...
- 11
1
vote
0
answers
100
views
singular integral operators
Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.
My ...
- 4,135
3
votes
1
answer
1k
views
Continuation (extension) of harmonic functions
Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
- 4,135
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
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
2
votes
0
answers
86
views
$L^2$-estimate for an elliptic equation
Given $(M, g)$ a compact Riemannian manifold of dimension $n \geq 3$, we consider the following equation for the function $\phi$:
$$
-\Delta \phi + R \phi + \tau^2 \phi^{N-1} = \frac{A^2}{\phi^{N+1}}
...
- 1,263
7
votes
1
answer
558
views
minimal surfaces in $S^n$
Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...
- 914
3
votes
2
answers
339
views
Converse to Lichnerowicz Vanishing Theorem?
The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...
- 787
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
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
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
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
13
votes
2
answers
816
views
$C^0$ estimate for solutions of elliptic PDE with Neumann BC
I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a $C^\...
- 131
5
votes
2
answers
459
views
A question on certain elliptic PDE
Consider the elliptic PDE
$$(CR)\;\;\;\;\;\;\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence $$(LAP)\;\;\;\;\;\;U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are ...
- 356
11
votes
3
answers
1k
views
How to define the square root of $1-\Delta $?
If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$?
The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential ...
- 1,441
5
votes
3
answers
1k
views
Elliptic theory on compact manifolds
Maybe this is silly.
On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation
$$ \Delta u=f \quad\text{ in $\Omega$}$$
$$ u=0\quad\text{ on $\partial\Omega$}.$$
One has the following ...
- 327
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
3
votes
2
answers
845
views
How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?
To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...
- 1,083