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Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
Joel Fine's user avatar
  • 6,247
15 votes
0 answers
1k views

Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
Marc Nardmann's user avatar
13 votes
0 answers
372 views

Finite dimensional approximation of Donaldson theory

In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
Chris Gerig's user avatar
  • 17.5k
12 votes
0 answers
478 views

What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...
Dan Ramras's user avatar
  • 8,803
11 votes
0 answers
364 views

Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...
Greg Zitelli's user avatar
  • 1,104
8 votes
0 answers
208 views

(Un)bounded Geometry and Sobolev Spaces

This post is related to this and this post. It is known that on a complete Riemannian manifold, the space $C^\infty_c(M)$ is generally not dense in the Sobolev spaces $W^{k, p}(M)$ ($1 \leq p < \...
Matthias Ludewig's user avatar
8 votes
0 answers
196 views

Parametrizing derivations from the algebra of smooth functions on a manifold to its dual

$\newcommand{\Der}{\operatorname{Der}}$ $\newcommand{\Real}{{\mathbb R}}$ (Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
Yemon Choi's user avatar
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7 votes
0 answers
80 views

Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator

Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
MyShepherd's user avatar
7 votes
0 answers
237 views

Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
geometricK's user avatar
  • 1,903
7 votes
0 answers
501 views

intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
R Mary's user avatar
  • 979
7 votes
0 answers
116 views

Bundles over Function Spaces

Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
elena's user avatar
  • 71
7 votes
0 answers
199 views

Central Extension of Continuous Loop Group

For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
Christoph Wockel's user avatar
6 votes
0 answers
318 views

Is there any connection between deformation theory in algebraic geometry and perturbation theory in functional analysis/PDEs?

Particularly, is there any connection between formal/first-order/infinitesimal deformation theory and perturbation theory? Both subjects involve "perturbing" some structure at a point, so ...
xuq01's user avatar
  • 1,094
6 votes
0 answers
532 views

Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
TheWildCat's user avatar
6 votes
0 answers
493 views

Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following: Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
uro's user avatar
  • 71
6 votes
0 answers
324 views

Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings, Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...
dominik's user avatar
  • 61
5 votes
0 answers
360 views

Injectivity of div–curl operator

$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system \begin{align*} Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\ u &= 0 \...
Chris's user avatar
  • 419
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 $...
Aleksandr Alekseev's user avatar
5 votes
0 answers
104 views

On the embedding of manifolds into infinite-dimensional spaces

Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
erz's user avatar
  • 5,529
5 votes
0 answers
211 views

Infinitesimal Generator of Billiard Flow

The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
Matthias Ludewig's user avatar
5 votes
0 answers
394 views

construction of heat kernels for non-compact manifolds with boundary

Recently, I am studying heat semigroup for noncompact manifolds with boundary. In Issac Chavel's book "eigenvalues in Riemannian geometry". "Given a noncompact Riemannian manifold, it need not be ...
wang mu's user avatar
  • 199
5 votes
0 answers
584 views

Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...
user avatar
4 votes
0 answers
126 views

Darboux integral for non-polynomial ODEs

Given a polynomial ODE in $n$-dimensions of maximal degree $d$ $$ \dot{x}_j=f_j(x)=\sum_{i_1,\dots,i_n=1}^d a_{i_1,\dots,i_n}^j x_1^{i_1}\dots x_n^{i_n} \quad \forall j=1,\ldots,n $$ we define ...
NicAG's user avatar
  • 247
4 votes
0 answers
334 views

Hodge decomposition on non-compact manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition $$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
B.Hueber's user avatar
  • 1,171
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?...
Jesús A. Álvarez López's user avatar
4 votes
0 answers
199 views

Spectral problems with the wrong sign on the Poincaré disk

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
Bilateral's user avatar
  • 2,816
4 votes
0 answers
382 views

Reference Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173

I have been searching without success for the reference: Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173 It is cited in many related works. In ...
E. Schulz's user avatar
  • 201
4 votes
0 answers
115 views

Delta distributions that are smooth on strata of a singular manifold

This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
Dmitry Vaintrob's user avatar
4 votes
0 answers
143 views

Sobolev space of maps between manifolds with boundary

Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds with non-empty smooth boundary. If we consider the Sobolev space $W^{1,p}(M,N)$, is there a reference on how to model this as a manifold? If ...
Somnath Basu's user avatar
  • 3,423
4 votes
0 answers
343 views

Diffeomorphism group action on the space of embeddings

Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...
Hang's user avatar
  • 2,789
4 votes
0 answers
120 views

Representation on square integrable sections of a principal bundle

Let $X\rightarrow Y$ be a smooth principal $G$-bundle for some Lie group $G$. Then $L^2(X)$ has a natural $G$-action determined by fibrewise action of $G$ on $X$. We have an abstract isomorphism of ...
geometricK's user avatar
  • 1,903
4 votes
0 answers
322 views

Cauchy-Riemann Operators and Selberg Zeta Function

The determinant of hyperbolic Maaß-Laplacian operator on arbitrary tensors and spinors can be written in terms of Selberg zeta function. Is there a corresponding formula for the determinant of the ...
QGravity's user avatar
  • 989
4 votes
0 answers
172 views

Donnelly-Fefferman growth of eigenfunctions

Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...
user91752's user avatar
4 votes
0 answers
315 views

Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...
K. P's user avatar
  • 51
4 votes
0 answers
241 views

"Partition" of a smooth function in $\mathbb R^2$

This is a question asking for reference. I have a proof of the following. Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions $f_1=...
PepeToro's user avatar
  • 231
3 votes
0 answers
108 views

A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
B.Hueber's user avatar
  • 1,171
3 votes
0 answers
153 views

Quasimode construction on a compact Riemannian manifold

Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
Y. Paka's user avatar
  • 131
3 votes
0 answers
110 views

On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
Pelle Steffens's user avatar
3 votes
0 answers
126 views

A path with zero increments and positive area

I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
Martin Geller's user avatar
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 ...
MyShepherd's user avatar
3 votes
0 answers
479 views

de Rham currents/distributions on manifolds with boundaries

My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
Bence Racskó's user avatar
3 votes
0 answers
289 views

Are smooth functions with compact support a core for the Laplacian on compact manifolds with boundary?

If $M$ is a complete Riemannian manifold and $L$ is the Friedrichs extension of the Laplacian $-\Delta$, then it is known (first proven by Gaffney in the '50) that $C_0 ^\infty (M)$ is a core for $L$. ...
Alex M.'s user avatar
  • 5,407
3 votes
0 answers
164 views

Extension of normal vector field to a domain

Let $\Omega \subset \mathbb R^3$ be a bounded regular simply connected domain contained in a ball $S$. Assume also that $\Omega$ is simply connected by surfaces (i.e. every regular closed surface ...
user934318's user avatar
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 $\partial M$....
Laithy's user avatar
  • 969
3 votes
0 answers
73 views

The diversity of Riemannian metrics adapted to a given foliation، A Krein Millman view point(2)

Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise ...
Ali Taghavi's user avatar
3 votes
0 answers
125 views

Green operator of elliptic differential operator and radius of convergence

Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
Simon Parker's user avatar
  • 1,383
3 votes
0 answers
58 views

Criteria for density of subgroup of diffeomorphism group

Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
ABIM's user avatar
  • 5,405
3 votes
0 answers
91 views

Pseudodifferential operator associated to a self-adjoint extension of a symmetric operator on an incomplete manifold

Let $D$ be the Dirac operator acting on a spinor bundle $S$ over a complete Riemannian manifold $M$. Then $D$ is an essentially self-adjoint operator on $L^2(S)$. Suppose there is a compact subset $K\...
geometricK's user avatar
  • 1,903
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(\...
geometricK's user avatar
  • 1,903
3 votes
0 answers
180 views

When is a minimal immersion holomorphic?

Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let: $\phi\colon X\to Y$ be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
Bilateral's user avatar
  • 2,816