-1
votes
0answers
74 views

If there is a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be hypersurfaces of dimension $n$ in $\mathbb{R}^{n+1}$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Denote the Laplace-Beltrami operator by $\Delta_{S(\cdot)}$. Let ...
6
votes
1answer
228 views

Are there nontrivial real functions of 2 real variables with gradient having constant euclidian norm on each level line?

Let $F$ be the class of locally Lipschitz continuous functions $z=f(x,y)$, from $\mathbb R \times\mathbb R \to\mathbb R,$ such that the euclidean norm $|\ \mathrm{grad}\ f (x,y)\ |$ of its gradient ...
0
votes
0answers
46 views

Guassian upper bound of the heat kernel implies ultracontrativity?

For spaces satisfies uniformlly local doubling and Poincare inequality (for example, Riemannian manifold with Ricci curvature bounded below RCD(K,N)). By Sturm's paper, we have bounds on the heat ...
1
vote
0answers
78 views

Buseman function for Riemanniam manifolds with two ends and $Ric\ge -(n-1)$ [closed]

It's well known that if M is a Riemannian manifold with $Ric \ge 0$ and contains a line $\gamma $. Set $${\gamma _ + } = \gamma | {_{[0, + \infty )}} ,{\gamma _ - } = \gamma | {_{[ - \infty ,0)}} ...
1
vote
1answer
172 views

Existence of positive solutions of a linear PDE on closed manifolds

I was wondering is there a sufficient condition (or sufficient and necessary condition) for the existence of positive solutions of the following linear PDE on a closed manifold $(M, g)$, ...
1
vote
1answer
136 views

A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...
0
votes
0answers
27 views

When is Dirichlet solution from disk to Riemannian manifold Holder continuous near the boundary?

$D$ is the two-dimensional unit disk. $X$ is a compact Riemannian manifold. Let$\phi \in W^{1,2}(D,X)$ and define $$ W^{1,2}_{\phi}=\{v \in W^{1,2}(D,X):Trace(v)=Trace(\phi)\} $$ Let $$ ...
11
votes
3answers
656 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 ...
0
votes
1answer
74 views

harmonic maps from cone to $S^2$ locally lipschitz?

Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
0
votes
0answers
89 views

Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)

Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
1
vote
0answers
114 views

Half-wave group $e^{it\sqrt{-\Delta_g}}$ for large $t$

Consider the Laplace-Beltrami operator $\Delta_g$ on compact Riemannian manifold $(M,g)$, then $e^{it\sqrt{-\Delta_g}}f$ is the solution of the following Cauchy problem. $$ ...
0
votes
0answers
68 views

Dirichlet integral of harmonic functions on manifold controlled by radius?

M is an n($\ge2$)-dim Riemannian manifold with Ricci curvature bounded below. $\Omega$ is a domain in M. $$\Delta u=0, x\in \Omega;u=f ,x\in \partial \Omega$$ f is Lipschitz. Is there a constant c ...
1
vote
2answers
131 views

How to show this integral on boundary of Lipschitz domain is finite?

Sorry for asking a basic question but this did not get answered on M.SE. Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that $$\int_{\partial\Omega} ...
1
vote
1answer
179 views

Harmonic function defined on a cone

It's well known that: Given a continuous function defined on the boundary of the disk, then there exists a unique harmonic function in the interior of the disk. What if we replace the disk by a cone? ...
1
vote
1answer
131 views

If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lipschitz hypersurface)?

Let $M$ be a bounded hypersurface. Let $f \in H^{\frac 12}(M)$ and let $\varphi\colon M \to \mathbb{R}$ be a Lipschitz function. When $M=\Omega \subset \mathbb{R}^n$ an open domain, we know that the ...
4
votes
0answers
256 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 ...
2
votes
1answer
196 views

Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
0
votes
0answers
62 views

$|\nabla f|^2 \in W^{1,2}(M)$ for harmonic function f on manifolds with two nonparabolic ends?

We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole M. ...
0
votes
0answers
133 views

Cheeger-Gromov-Taylor theory on manifolds with boundary

I was reading the paper "Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds" by Cheeger, Gromov and Taylor and I am ...
1
vote
2answers
237 views

How to construct a harmonic function with non-zero gradient on manifold with two nonparabolic ends?

We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole $M$. More ...
2
votes
0answers
124 views

How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...
4
votes
1answer
262 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 ...
2
votes
1answer
238 views

Fully non-linear PDE

A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...
13
votes
5answers
901 views

Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
4
votes
3answers
404 views

Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question: "Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?" I believe the answer to ...
2
votes
1answer
253 views

Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...
2
votes
1answer
221 views

unique continuation property for overdetermined elliptic PDE

On a closed manifold $M$, let $P(f)=0$ be a linear overdetermined elliptic system of PDE of 2nd order with smooth coefficients. By overdetermined ellipticity, I mean the principal sympbol is ...
2
votes
2answers
248 views

worst regularity of f ensuring u is locally Lipschitz for $\Delta u = f$

Assume $M$ is a Riemannian manifold, and $\Omega $ is a bounded domain. Consider the Poisson equation: $$\Delta u = f \qquad \text{with }u \in {W^{1,2}}$$ What is the worst regularity of $f$ which ...
2
votes
1answer
89 views

Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator?

Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow $$\int_{\mathbb{R}^{n}} K_{t}(x, \cdot)\,dy = ...
1
vote
1answer
104 views

metric scaling for an inequality

I read a lemma 1.12 in Tobias H.Colding's paper "Ricci curvature and volume convergence"."Suppose that $Ri{c_{{M^n}}} \ge \left( {n - 1} \right)\Lambda {R^{ - 2}}$,p and $q \in M$ with d(p,q)>8R,and ...
3
votes
2answers
144 views

Boundedness of Solutions to $\Delta u = f u$ on $R^2$

Consider the Laplacian $\Delta = d/dx^2 + d/dy^2$ on $\mathbb{R}^2$. This is true: Let $f$ be a nonnegative function, not identically zero. Then any positive solution of $\Delta u = f u$ is ...
0
votes
2answers
226 views

Estimates for Green's function

Let $n$ - dimension $\geq 3$. Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < ...
3
votes
2answers
217 views

Analytic dependence on the metric

It is often used implicitly that the maps which associate to metrics curvature quantities (Riemann, Ricci, scalar curvature) and Differential operators like the Laplacian are analytic maps between ...
8
votes
4answers
720 views

Is there anyway to rewrite a partial differential equation using language of differential forms, tensors, etc?

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there anyway (or any algorithm) ...
0
votes
0answers
113 views

Gradient estimates for subsolutions of elliptic equations

Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and $\Delta u + \lambda u = 0,$ where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's ...
0
votes
0answers
102 views

Sobolev spaces on hypersurfaces

I am learning about Sobolev spaces on hypersurfaces. Let $S$ be a $C^k$-hypersurface with boundary for some $k$. In order to define a weak derivative, one needs $k \geq 2$ because the integration by ...
2
votes
1answer
138 views

Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...
3
votes
1answer
210 views

What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
7
votes
1answer
322 views

Representing immersions from a surface into 3-space

Let $\mathbb T^2=(S^1)^2$ be the 2-torus, for convenience. $\def\Imm{\operatorname{Imm}}$ Consider the Frechet manifold of immersion $\Imm(\mathbb T^2, \mathbb R^3)$ and the smooth mapping ...
2
votes
1answer
224 views

Trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold. ...
2
votes
1answer
212 views

Is there a lower bound for variance in terms of curvature?

If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero ...
1
vote
2answers
514 views

$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?

Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
3
votes
1answer
426 views

Density of smooth functions in Sobolev spaces on manifolds

Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have ...
3
votes
1answer
337 views

conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...
0
votes
1answer
67 views

Integral of a harmonic function on a manifold with two non-parabolic ends

Let M be a complete Riemannian manifold.Suppose there are two non-parabolic ends on M with respect to $M\backslash {B_p}\left( {{R_0}} \right)$Then there is a harmonic function f on M.Is it right that ...
10
votes
1answer
436 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...
1
vote
0answers
153 views

multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ , is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $ for $\left | \sigma \right |\leqslant ...
2
votes
1answer
130 views

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?
0
votes
1answer
314 views

Existence of a function

[also asked here http://math.stackexchange.com/questions/307197] All arguments are in $\mathbb{R}^3$. Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a compact set $\Omega$. ...
0
votes
0answers
183 views

Eigenfunctions of the laplace on the 2-sphere with conformal metric induced by schwarzschild

Hi, it's well known that the coordinates $x_1$, $x_2$, $x_3$ are the first three eigenfunctions with positiv eigenvalues ($=2/r^2$) of the negative laplace-beltrami operator ${-}\triangle$ on the ...