# Tagged Questions

**3**

votes

**2**answers

152 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**

vote

**2**answers

320 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 ...

**14**

votes

**1**answer

374 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 ...

**2**

votes

**2**answers

348 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**

vote

**2**answers

822 views

### Geometric Mean Value Property

Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...