8
votes
1answer
209 views

Dimension of eigenspaces of Laplacian on a compact Riemannian manifold

Let $M$ be a compact smooth manifold, let $g$ a riemannian metric and let $\Delta_{g}$ the Laplacian operator on functions induced by $g$. Is there a (topological?) bound on the dimension of $n$-th ...
0
votes
1answer
321 views

Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)? For Hartogs's Extension Theorem see here: ...
0
votes
1answer
152 views

Flat connection, finite-dimensional space of covariant constant one forms

hallo, I have the following question: Let $U\subset \mathbb{R}^{n}$ be an open subset. Furthermore, let $\nabla$ be a flat connection on $U$ (not necessary Levi-Civita). How can one show that the ...
2
votes
1answer
305 views

Constructing the imaginary part of a holomorphic function

Hallo, Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function ...
3
votes
2answers
422 views

Corona Theorem in several variables

Hallo, I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...
1
vote
0answers
173 views

HyperKaehler manifolds are Ricci-flat

Hi, I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = ...
2
votes
2answers
315 views

Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold

Hallo, I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, ...
0
votes
1answer
211 views

Unique symplectic form in an adapted complex structure

Hallo, I ave the following question: Due to Stenzel, Lempert, Szöke ect. we know that a Riemannian manifold $(M,g)$ admits a complex structure on an neighbourhood of the cotangent bundle. This ...
4
votes
0answers
187 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. ...
-2
votes
1answer
312 views

holomorphic extension of a function [closed]

hi, I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : ...
1
vote
1answer
203 views

Kähler manifold with Ricci-flat Kähler form

hallo, I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
2
votes
0answers
339 views

Off-diagonal asymptotic expansion of the Bergman kernel on hyperbolic Riemann surfaces

Let $X$ be a compact Riemann surface of genus at least 2. Let $K$ denote the canonical line bundle, and $E$ be any vector bundle. Let $P^{(m)}$ be the projection map from the space $L^2(X,K^mE)$ of ...
17
votes
0answers
582 views

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 ...
10
votes
2answers
511 views

A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...