All Questions
18 questions
15
votes
2
answers
888
views
Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?
Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms
$$
\Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
- 35.5k
14
votes
1
answer
668
views
Why are we interested in spectral gaps for Laplacian operators
Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
- 185
6
votes
3
answers
3k
views
Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?
The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
Is there another definition?
For example, for ...
- 4,312
5
votes
1
answer
932
views
Mellin transform between heat kernel and zeta-function
For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of square-...
- 617
5
votes
1
answer
449
views
Practical way to check whether a distribution is conormal
Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 \...
- 1,329
5
votes
0
answers
191
views
Index of the Fredholm operator
I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
4
votes
1
answer
345
views
Differential Operators On A Curve And On Osculating Circle
Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
- 143
4
votes
1
answer
474
views
Are these operators defined on 2D surfaces self-adjoint?
My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether ...
- 199
4
votes
1
answer
311
views
Conormal distributions and the wave front set
Let $X$ be a smooth closed manifold and $Y$ a regular submanifold. For all conormal distributions at $Y$ on $X$, their wave front set is contained in the conormal bundle of $Y$. Is the reciprocal true?...
3
votes
0
answers
198
views
Dirichlet to Neumann operator and the Riesz transform
Consider the manifold $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $\gamma$ be the induced metric on $\partial M$....
- 969
3
votes
0
answers
53
views
Controlling a Schwartz kernel near the diagonal
Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
- 1,903
2
votes
2
answers
162
views
Finding a specific Global Smooth Function
Any help with this problem would be appreciated. Thanks
Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...
- 4,135
2
votes
0
answers
524
views
What essential property justifies the name "derivative"?
Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
- 643
1
vote
1
answer
547
views
de Rahm Laplace operator on forms bounded
Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The ...
- 11
1
vote
1
answer
161
views
Continuity of image of resolvent operator with respect to resolvent parameter
Suppose $D$ is a first-order differential operator on a manifold $M$ and that the inverse $(D+t)^{-1}:H^0(M)\rightarrow H^1(M)$ exists for all $t > 0$, where $H^i(M)$ is the $i^\text{th}$ Sobolev ...
- 1,903
1
vote
0
answers
235
views
Fredholmness of elliptic operator on Hölder spaces
Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
- 163
0
votes
0
answers
141
views
The tensor product of two Fredholm operators
What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
- 356
0
votes
0
answers
76
views
Linear dependence of the derivatives of a vector valued function
Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function
$$
g:\mathbb{R}^5\rightarrow\mathbb{R}^5
$$
given by
$$
g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
- 8,998