All Questions
Tagged with riemannian-geometry elliptic-pde
101 questions
6
votes
1
answer
206
views
Reference for $\epsilon$-regularity
I am looking for a reference for the following $\epsilon$-regularity statement: let
$(M,g)$ be a Riemannian manifold of dimension $n$,
$\Delta=dd^*+d^*d$,
$B_r$ denotes a ball of radius $r$ around a ...
- 954
6
votes
1
answer
196
views
On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
- 1,429
3
votes
0
answers
219
views
Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
- 6,155
2
votes
0
answers
56
views
Convergence of conformal metrics with prescribed curvature
We know that for any function $K: \mathbb{D} \to \left[-a, -b\right]$, where $a, b > 0$, there is a unique metric $h$ on the disk $\mathbb{D}$ which is conformal to $dz^{2}$, and has curvature ...
- 63
2
votes
0
answers
57
views
A sequence of conformal metrics with bounded negative curvatures on the disc
Let $\mathbb{D}$ denote the unit disk, and let $h_{-1}$ be the unique hyperbolic metric on $\mathbb{D}$ which is conformal to $dz^{2}$.
Take a sequence of smooth complete metrics $h_{n} = e^{\rho_{n}} ...
- 63
5
votes
0
answers
277
views
Elliptic equation on differential forms
Let $\Sigma$ be an $n$-dimensional smooth closed manifold ($n\ge 3$) with a non-continuous metric $g\in W^{2,\frac{n}2}\cap L^{\infty}(\Sigma)$. Let $g'$ be a fixed smooth metric on $\Sigma$, there ...
- 435
5
votes
1
answer
463
views
Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
- 1,171
7
votes
2
answers
626
views
Elliptic regularity on manifolds: Is this true?
Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
- 1,171
3
votes
0
answers
173
views
$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature
This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ .
The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
- 155
8
votes
1
answer
357
views
Estimates of $\Delta|\nabla u|$ for harmonic function $u$
The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$$
\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
- 155
3
votes
1
answer
331
views
Does the strong maximum principle for minimal surfaces hold in Riemannian manifolds?
In Euclidean spaces, the following maximum principle for minimal surfaces are well known.
Theorem: If $\Sigma_1$, $\Sigma_2 \subset \mathbb{R}^n$ are complete connected minimal hypersurfaces, $\...
- 438
1
vote
0
answers
110
views
Moser iteration epsilon-regularity for non-linear system in general dimension
I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
3
votes
0
answers
82
views
Dirichlet-to-Neumann map is analytic
Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
- 2,095
2
votes
0
answers
94
views
Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics
Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
- 2,095
4
votes
0
answers
129
views
Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
user516086
1
vote
1
answer
256
views
Moser iteration in dimension $6$
Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying
\begin{align*}
\Delta f \leq gf-\frac{3}{4}f^2
\end{align*}
Where $g$ is another smooth ...
- 954
3
votes
0
answers
62
views
Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field
Setting
Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
- 808
1
vote
1
answer
90
views
Infimum of the normalized Laplacian eigenvalues
Let $(M^n,g)$ be a compact Riemannian manifold. The spectrum of the Laplacian operator $\Delta_g = -\operatorname{div} \nabla$ consists of an increasing and diverging sequence of positive eigenvalues:
...
- 2,095
2
votes
0
answers
80
views
$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow
Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,...
- 1,219
2
votes
1
answer
172
views
Maximizing the first Neumann eigenvalue on disks
Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that
$$\mu_1(g) \operatorname{...
- 2,095
4
votes
1
answer
193
views
Approximate isometric embeddings of surfaces
The fundamental theorem of surfaces states that if symmetric matrices $g_{ij}$, $l_{ij}\colon U\subset R^2\to R$, where $U$ is open and $g_{ij}$ is positive definite satisfy the Gauss and Codazzi ...
- 7,149
4
votes
0
answers
137
views
Eigenvalues of Schrödinger operator with Robin condition on the boundary
Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
- 2,095
2
votes
0
answers
134
views
Critical points of a strictly subharmonic function
Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant:
\begin{equation}
\Delta u = A > 0....
- 5,038
5
votes
3
answers
620
views
Poisson equation on manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...
- 1,171
1
vote
0
answers
86
views
Poisson equations for tensors on compact Riemannian manifold
Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation
$$\Delta f=S$$
where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...
- 1,171
4
votes
1
answer
377
views
The minimal surface equation in a Riemannian metric
Let $\Omega \subset \mathbf{R}^2$ be a domain, and let the cylinder $\Omega \times \mathbf{R}$ above it be endowed with a Riemannian metric $g$. (Note this is not assumed invariant in the vertical ...
- 5,038
7
votes
1
answer
514
views
Eigenvalues of the Laplacian on surfaces with boundary
Let $\Sigma$ be a compact smooth surface with boundary. Is it true that the supremum
$$\sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ smooth Riemannian metric on $\Sigma$} \}$$
...
- 2,095
5
votes
1
answer
359
views
Sobolev embedding on sphere
Let $S$ be a two-dimensional sphere, $\Delta$ be the Laplace-Beltrami operator on $S$ and $L^p(S)$, $p\geq 1$, be the usual $L^p$ space of real-valued functions on $S$. We also set $\|f\|_{H^\alpha(S)}...
- 552
0
votes
0
answers
57
views
If a Dirichlet problem is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?
If a Dirichlet problem (elliptic PDE, in $R^{n}$) is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?
For example, $B$ is the unit ball in $R^{n}$, the ...
- 809
2
votes
0
answers
93
views
Elliptic regularity estimate with robin boundary condition in Yamabe problem on manifolds with boundary
I meet the following boundary problem in Escobar's Yamabe Problem On Manifolds With Boundary.https://projecteuclid.org/journals/journal-of-differential-geometry/volume-35/issue-1/The-Yamabe-problem-on-...
- 217
1
vote
1
answer
103
views
How to calculate the infimum of Yamabe functional on upper hemisphere
We introduce the following functional to study Yamabe problem with boundary.
$$
Q_g(\varphi)=\frac{\int_M\left(|\nabla \varphi|_g^2+\frac{n-2}{4(n-1)} R_g \varphi^2\right) d v+\frac{n-2}{2} \int_{\...
- 217
2
votes
0
answers
103
views
Question about the second order linear elliptic PDE on closed manifold
Recently I see a question
linear second order PDE
in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE
$$
\Delta u +B^{i}(x)u_{i}+C(x)u=f
$$
is ...
- 809
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
1
vote
1
answer
288
views
A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used
I'm reading Tawfik - The Yamabe problem: the PDE is
$$
\Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{q-1}. \label{1}\tag{1}
$$
Theorem (Yamabe). For $2<q<N=N=2 n /(n-2)$, there exists a $C^{\...
- 809
4
votes
0
answers
82
views
On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
- 4,135
2
votes
0
answers
85
views
Proving an eigenvalue bound without resorting to Weyl's law
Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
- 4,135
2
votes
0
answers
77
views
Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator
I would appreciate any answers or even references for the following problem.
Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
- 4,135
5
votes
0
answers
101
views
When are nodal lines on a sphere geodesics?
Let $(S^2, g)$ be a Riemannian sphere and let $L := \Delta_{S^2} + q$ be a Schrödinger operator on $S^2$. Suppose that $L$ has index equal to one and that $u \in C^{\infty}(S^2)$ ($u \neq 0$) lies in ...
- 2,095
1
vote
1
answer
75
views
A property for generic pairs of functions and metrics
Let $M$ be a compact smooth manifold with a smooth boundary. Given a smooth Riemannian metric $g$ on $M$, we denote by $\{\phi_k\}_{k=1}^{\infty}$ an $L^2(M)$--orthonormal basis consisting of ...
- 4,135
2
votes
1
answer
397
views
Fractional Laplacian on closed manifolds
Naturally given any $s\in (0,1)$, the fractional Laplacian, $(-\Delta_g)^s u$ on a closed Riemannian manifold can be defined via spectral decomposition of $-\Delta_g$. There is another formulation of ...
- 4,135
1
vote
0
answers
98
views
Gap phenomenon vs Rigidity results for surfaces
I am trying to understand the differences between the rigidity results and gap results for a given surface immersed into some manifold. For instance, a Gap theorem proved here (Theorem 2.7) says ...
- 11
2
votes
0
answers
208
views
Can you compute one eigenspace without computing them all?
Maybe the simplest non-trivial settings in which the spectrum of the Laplacian be can be computed is on the round sphere $\mathbf{S}^n$, and for products of manifolds. I want to use the two as ...
- 5,038
1
vote
0
answers
86
views
Hypoellipticity of a heat-like parabolic operator on Riemannian manifolds - reference request
Let $(M,g)$ be a Riemannian manifold and $L$ be a differential operator on $M$, with smooth coefficients, such that its symbol be $g$ (a "generalized Laplacian").
Where can I find proved ...
- 5,407
4
votes
1
answer
337
views
Alternative to well-known trace estimate in Riemannian geometry?
Let $g,\hat{g}$ be two Riemannian metrics with volume forms $dv_g$, $dv_{\hat{g}}$, respectively. A standard estimate in the subject is the following: $$\text{tr}_g(\hat{g}) \leq \text{tr}_{\hat{g}} (...
user105074
3
votes
0
answers
101
views
Minimal normal graph
Let $(M^3,g)$ be a complete and orientable Riemannian $3$-manifold and let $\Sigma^2 \subset M$ be a compact orientable totally geodesic surface embedded in $M$. For $f \in C^{2,\alpha}(\Sigma)$ with ...
- 2,095
3
votes
0
answers
142
views
Homogeneous Carnot group, its Lie algebra and Carnot-Carathéodory ball
Background: Let the smooth vector fields $X=(X_1,\cdots,X_m)$ define on $\mathbb{R}^n$ and they satisfy the following assumption:
(H1): There is a dilation structure
$$\delta_{t}:\mathbb{R}^n\to \...
- 561
2
votes
0
answers
144
views
Why are products of spheres integrable?
Let $n+1 \geq 3$ be an integer and $p + q = n$. Inside the unit sphere $\mathbf{S}^{n+1}$ the product
\begin{equation}
\mathbf{S}^p(\sqrt{p/n}) \times \mathbf{S}^q(\sqrt{q/n}) = \{ (X,Y) \in \mathbf{R}...
- 5,038
5
votes
0
answers
199
views
Differential equation on a Riemannian manifold
Let $(M,\mu)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\phi+h$, the decomposition of $\theta$ according to the Hodge decomposition. ...
5
votes
2
answers
376
views
Flapping wings: on a question of Kapouleas
The Lawson minimal surfaces $\xi_{1,g} \subset \mathbf{S}^3$ are minimal surfaces with genus $g$. In Lawson's original construction [Law70]
these were constructed from geodesic triangulations. An ...
- 5,038
3
votes
0
answers
65
views
Elliptic equations in semi-infinite strips
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
- 4,135