# Tagged Questions

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40 views

### Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...

**0**

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**1**answer

86 views

### Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form
$$|u(t)|_{L^p} \leq ...

**2**

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**0**answers

105 views

### Variational inequality on Manifold

Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla ...

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**0**answers

142 views

### An inequality on closed manifolds

Excuse me again, I am not sure what kind of question the following is. Any suggestion is appreciated!
Let $(M,g)$ be a closed Riemannian manifold, $g$ is real analytic. Let $u,v$ be two nonnegative ...

**1**

vote

**1**answer

187 views

### Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function
$h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is:
...

**2**

votes

**1**answer

258 views

### A question on Schrodinger operator

I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated!
Let $(M,g)$ be ...

**4**

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**0**answers

79 views

### Reference for short time existence of paraobolic PDE on bundles

I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...

**3**

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**0**answers

110 views

### $\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$.
Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...

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77 views

### A question on the maximum principle for Schrodinger operators

Let $(M^n,g)$ be a closed Riemannian manifold, and $L=-\Delta+V$ be a Schrodinger operator, $V\in C^{\infty}(M)$. In answers to the two questions (First eigenvalue of Schrödinger operator is simple 1 ...

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**0**answers

98 views

### Estimate the smallest eigenvalue of a Schrodinger operator

There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of Grigor'yan-Nadirashvili-Sire and references therein. I wonder how to ...

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**1**answer

344 views

### Monge–Ampère type equation

Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''_{xx}\leq 0, B''_{yy}\leq 0$ (i.e. it is bicocncave function).
I am looking for the solutions among of ...

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**1**answer

125 views

### Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?

Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.
Assume that $S$ satisfies ...

**4**

votes

**1**answer

467 views

### The space of diffeomorphisms on a manifold

It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions ...

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**1**answer

102 views

### metric has morse index 2

I am reading Richard Schoen's classical example on the multiplicity of solutions of yamabe problem. He says on $S^1(T)\times S^{n-1}$, there exists a critical number $T_0$ such that if $T\leq T_0$, ...

**3**

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**1**answer

243 views

### Derivation of yamabe flow

I am reading papers about yamabe flow. I have a problem about how people derive it as a gradient flow.
Suppose we have $(M,g_0)$, $g(t)=u^{\frac{4}{n-2}}(t)g_0$ is another conformal metric. Let ...

**6**

votes

**1**answer

239 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 ...

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**0**answers

55 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 ...

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**0**answers

84 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)}} ...

**4**

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**1**answer

300 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 $(M, g)$,
...

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**1**answer

231 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 ...

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**0**answers

59 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

**3**answers

699 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 ...

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**1**answer

75 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?

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103 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 ...

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**0**answers

117 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.
$$
...

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74 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 ...

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**2**answers

141 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} ...

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**1**answer

197 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?
...

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**1**answer

137 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 ...

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**0**answers

266 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 ...

**3**

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**1**answer

219 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 $0$ ...

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65 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.
...

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137 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 ...

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**2**answers

254 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 ...

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125 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

**1**answer

280 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

**1**answer

246 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

**5**answers

961 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

**3**answers

438 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

**1**answer

279 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

**1**answer

267 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 ...

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votes

**2**answers

250 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

**1**answer

91 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 = ...

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**1**answer

109 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 ...

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**2**answers

145 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 ...

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votes

**2**answers

240 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

**2**answers

225 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 ...

**9**

votes

**4**answers

804 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) ...

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**0**answers

128 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 ...

**2**

votes

**1**answer

150 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 ...