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21 votes
0 answers
876 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 ...
Joel Fine's user avatar
  • 6,247
18 votes
1 answer
3k views

Theorem of Bryant in higher dimensions

I have the following question. I read about Bryant's theorem which says that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically embedded as ...
gary's user avatar
  • 221
18 votes
1 answer
951 views

Poincaré metric on the Riemann sphere minus more than two points

If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let ...
Amin Talebi's user avatar
13 votes
4 answers
3k views

Calabi - Yau Manifolds

I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one ...
J Verma's user avatar
  • 3,218
13 votes
1 answer
493 views

Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric

Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$. Assume there is a diffeomorphism $\nu:M\to N$ ...
user avatar
11 votes
1 answer
1k views

Solutions of equations characterizing a complex structure

Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $...
Amir Baghban's user avatar
9 votes
1 answer
407 views

Almost Complex manifolds of constant curvature

Edited (after R. Bryant comment) Let $(M,\cal J,g)$ be a almost Hermitian manifold (not necessary integrable). i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}...
C.F.G's user avatar
  • 4,195
9 votes
0 answers
283 views

Hermitian sectional curvature

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair. ...
seub's user avatar
  • 1,347
9 votes
0 answers
344 views

Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows: A) Is there a classification of the possible homeomorphism types of ...
Bilateral's user avatar
  • 2,816
8 votes
3 answers
1k views

Examples of manifolds that do not admit scalar flat metrics

The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories: (A) Every (smooth) function is a scalar curvature. (B) The manifold is strongly ...
Ryan Unger's user avatar
8 votes
2 answers
2k views

Kähler metrics for projective space that are not the Fubini-Study metric

For projective $N$-space $CP^{N}$, there is a canonical Kähler metric called the Fubini-Study metric. Do there exist other Kähler metrics for $CP^N$. If so, is there any classification of such metrics?...
Dyke Acland's user avatar
  • 1,479
8 votes
2 answers
695 views

Kronheimer's results on ALE spaces as hyperkahler quotients

Background: In his two papers from late 80s Kronheimer proved that any 4-dimensional ALE space is given by a hyperkahler quotient, say $X_{{\zeta_\mathbb{R}},{\zeta_\mathbb{C}}}(Q)$ where Q is a ...
Filip's user avatar
  • 1,677
8 votes
1 answer
297 views

''Are Hermitian metric pullbacks automatically via biholomorphisms?''

The awkward title is an attempt at approximating the following specific question: Let $(M^{2n}, J)$ be a complex manifold, suppose $g_0$ is a Riemannian metric $M$ compatible with $J$, and suppose $\...
Thisquestionisreallyhard's user avatar
8 votes
2 answers
453 views

Vector field with constant divergence around embedded submanifold

Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$. Question: Does there ...
StanleyT's user avatar
8 votes
1 answer
256 views

Do all symmetries of a Kähler quotient come from the original space?

For a Kähler manifold $M$, let $\operatorname{Iso}_{\mathbb{C}}(M)$ denote the group of holomorphic isometries. Suppose that $K$ is a compact subgroup of $\operatorname{Iso}_{\mathbb{C}}(M)$ and ...
user116804's user avatar
8 votes
1 answer
458 views

Different complexifications of a real analytic Riemannian manifold

I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well known fact that in a neighbourhood $U$ of the zero ...
Dmitri's user avatar
  • 101
7 votes
1 answer
607 views

Kähler metric with two compatible complex structures

Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$. Can we prove that $(M,g)$ is ...
Zhiqiang's user avatar
  • 891
7 votes
2 answers
436 views

Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
Nuxil's user avatar
  • 73
7 votes
2 answers
243 views

Length of simple closed curve in half-translation surface

Let $R$ be a Riemann surface of genus $g\ge 2$ and $q$ an holomorphic quadratic differential on $R$. Together they determine a semi-translation structure: an atlas on $X$ such that its changes of ...
User28341's user avatar
  • 609
7 votes
0 answers
656 views

Least area minimal hypersurface of $\mathbb C P^{n+1}$

After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was ...
Renato G. Bettiol's user avatar
6 votes
3 answers
2k views

What is the Weitzenböck formula for the $\bar\partial$-Laplacian

It is well-known that the Weitzenböck formula for the real Laplacian is $$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$ where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the ...
user31034's user avatar
6 votes
2 answers
706 views

Reference request: uniformization theorem

I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level. Any good powerpoint notes, short ...
TypoSpeed23's user avatar
6 votes
1 answer
338 views

Atiyah-Singer for Riemannian and Kaehler manifolds

I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
Dick Johnson's user avatar
6 votes
1 answer
392 views

Analytic representatives for Kahler classes

If we are given compact complex manifold $X$ and a Kahler class $[\omega]$, can we always find a positive definite representative $\omega \in [\omega]$ that is real analytic?
Italo's user avatar
  • 1,727
6 votes
1 answer
1k views

Holonomy of a Kähler manifold

Hi, I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla \...
Mina's user avatar
  • 93
6 votes
1 answer
881 views

Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...
user0029's user avatar
  • 109
6 votes
0 answers
144 views

What does it mean for the torsion to blow up?

Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian: Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
GradStudent's user avatar
6 votes
0 answers
397 views

Complex Riemannian metrics over real manifolds

There is a huge literature on complex manifolds and natural metrics over them, but I was unable to find references about Riemannian complex metric on real manifolds (for which we complexify tangent ...
Phil-W's user avatar
  • 1,035
5 votes
1 answer
1k views

On the complexification of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
Amir Baghban's user avatar
5 votes
1 answer
431 views

Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach

I'm analyzing the following isometric immersion of $(\mathbb H^2,g_D)$ in $(\ell^2,g_\infty)$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with \begin{align}\label{5.1} x_{2m-1}=\color{...
Zaragosa's user avatar
  • 143
5 votes
1 answer
389 views

Lengths of closed geodesics on a flat vs hyperbolic punctured torus

Let $T$ be a torus (oriented closed surface of genus 1), $p\in T$, and $T^* := T - \{p\}$. Let $\mu$ denote a flat structure on $T$. This can be obtained for example by choosing a uniformization $p_f:...
Will Chen's user avatar
  • 10.7k
5 votes
1 answer
515 views

Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space". The tangent space at ...
user4887's user avatar
5 votes
0 answers
131 views

Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?

The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...
AmorFati's user avatar
  • 1,379
5 votes
0 answers
173 views

reference for the weak compactness of currents

I am trying to follow the arguments in page 22 of the following paper k\"{a}hler currents and null loci It quotes the weak compactness of currents, I wonder if there is any reference about it. My ...
zach's user avatar
  • 151
5 votes
0 answers
148 views

Toponogov comparison theorem for complex manifold

I would like to know some reference for the Toponogov comparison theorem for complex manifolds, in particular for complex manifolds with bounded holomorphic sectional curvature. As far as I know, the ...
Daniel's user avatar
  • 303
4 votes
2 answers
575 views

Do transvers foliations induce complex structure?

Hallo, I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...
Marin's user avatar
  • 41
4 votes
1 answer
467 views

Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold

Hallo, I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ,...
bernard's user avatar
  • 53
4 votes
1 answer
151 views

Orientation-preserving isometric involution on compact Kähler manifold

Let $M$ be a compact Kähler manifold. If $\phi:M\to M$ is an orientation-preserving isometric involution does it have to be either holomorphic or anti-holomorphic?
user avatar
4 votes
2 answers
626 views

Uniqueness of Kähler form with same volume

Hallo, Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure $...
hapchiu's user avatar
  • 339
4 votes
1 answer
215 views

Examples of surfaces with negative Kahler curvature operator

Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator. Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces ...
Vamsi's user avatar
  • 3,383
4 votes
1 answer
337 views

Alternative to well-known trace estimate in Riemannian geometry?

Let $g,\hat{g}$ be two Riemannian metrics with volume forms $dv_g$, $dv_{\hat{g}}$, respectively. A standard estimate in the subject is the following: $$\text{tr}_g(\hat{g}) \leq \text{tr}_{\hat{g}} (...
user avatar
4 votes
0 answers
274 views

How many ways are there to characterise $\mathbb{P}^n$?

Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\...
AmorFati's user avatar
  • 1,379
4 votes
0 answers
104 views

Non-isomorphic compact Kähler manifolds not containing submanifolds biholomorphic to their conjugates

Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$. Assume there is a diffeomorphism $\nu:M\to N$ ...
user avatar
3 votes
1 answer
354 views

Holomorphic structures for line bundles over projective manifolds

Let $M$ be a compact K\"ahler manifold, which is assumed to be projective, i.e. there exists an ample line bundle over $M$ giving an embedding into $\mathbb{C}P^n$. Let $\mathcal{L}$ be a smooth line ...
Max Schattman's user avatar
3 votes
2 answers
657 views

Reference for Almost-Kahler geometry

Is there any comprehensive reference for Almos-Kahler geometry or more generally to Almost- Hermitian geometry ?
Hamed's user avatar
  • 1,236
3 votes
1 answer
505 views

non-existence of global coordinates

Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
Wakabaloola's user avatar
3 votes
3 answers
1k views

Rotation in Hyperkähler manifolds

Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...
hapchiu's user avatar
  • 339
3 votes
1 answer
426 views

Restriction of the Levi-Civita Connection to a Connection on the (Anti-)Holomorphic Forms

For a Riemannian manifold $(M,g)$, that is also a complex manifold, when does the Levi-Civita $\nabla_g$ connection restrict to a connection on the holomorphic forms $\Omega^{(\cdot,0)}$, and when ...
John McCarthy's user avatar
3 votes
1 answer
325 views

Geometric Morse theory ( and its complex analogy)

In the literature are there some concept of geometric version of Morse or Picard Lefschets theory? That is the comparison of level sets as Riemannian submanifold not merely as ...
Ali Taghavi's user avatar
3 votes
1 answer
388 views

Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold

Hallo, It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...
hapchiu's user avatar
  • 339