All Questions
94 questions
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 ...
- 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 ...
- 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 ...
- 183
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 ...
- 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 $\nu:M\to N$ ...
user164740
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 $...
- 601
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}...
- 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.
...
- 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 ...
- 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 ...
- 738
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?...
- 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 ...
- 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 $\...
8
votes
2
answers
452
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 ...
- 81
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 ...
- 81
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 ...
- 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 ...
- 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....
- 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 ...
- 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 ...
- 5,891
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 ...
- 93
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 ...
- 63
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 ...
- 203
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?
- 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 \...
- 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 ...
- 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 ...
- 651
6
votes
0
answers
396
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 ...
- 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 $...
- 601
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{...
- 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:...
- 10.7k
5
votes
1
answer
514
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 ...
- 53
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 ...
- 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 ...
- 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 ...
- 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 ...
- 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 ,...
- 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?
user164740
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 $...
- 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 ...
- 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}} (...
user105074
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 $\...
- 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 $\nu:M\to N$ ...
user164740
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 ...
- 993
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 ?
- 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....
- 277
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) ...
- 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 ...
- 1,462
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 ...
- 356
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 ...
- 339