All Questions
94 questions
2
votes
0
answers
34
views
Do internal stable sets contain big manifolds?
Given two strictly concave functions $u_{i}$ with continuous derivatives in $\mathbb{R}^{k}$. We define their upper levels at a point $x$ of these functions as the set of points y such that $u_i(y)>...
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
1
vote
0
answers
307
views
Fefferman metric and Einstein metric
From Lee's paper The Fefferman Metric and Pseudo hermitian Invariants, corresponding to any 3 dimensional strictly pseudo convex CR structure, there is a conformal class of Lorentzian metrics which ...
- 99
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
2
votes
1
answer
427
views
Is Thierry Aubin’s theorem true on Hermitian manifolds?
A classical theorem of Thierry Aubin states that:
Theorem (Aubin, T. 1979): If the Ricci curvature of a compact Riemannian manifold is
non-negative and positive at a point, then the manifold ...
- 4,195
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
3
votes
0
answers
165
views
Is a non vanishing holomorphic vector field necessarily a geodesible vector field?
Motivated by the "The obvious Fact" part of this answer,, we ask the following question:
First we recall a definition, which is used in the above link:
Definition: A non vanishing vector ...
- 356
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 $\...
0
votes
2
answers
553
views
A relation between gradient vector field and Hamiltonian vector field
Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.
Is there a Riemannian ...
- 356
2
votes
1
answer
590
views
Ricci form is closed?
Let $(M,g,J)$ be an almost Kähler manifold and let $\rho$ denote its Ricci form
$$
\rho(X,Y) = \operatorname{ric}^{\mbox{c}}(JX,Y)
$$
where $\operatorname{ric}^{\mbox{c}}$ is the $J$-invariant part of ...
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
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
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
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
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
2
votes
0
answers
269
views
Is a G-invariant metric always Kähler-Einstein?
Suppose there is a Hermitian symmetric space of compact type $X$. It is realized in the following way: $X\hookrightarrow\mathbb{P}^N$ and equipped with the induced Fubini-Study metric $g$.
What's ...
- 1,121
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
1
vote
3
answers
572
views
Special connection of vector bundle over real manifold
Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...
- 131
1
vote
0
answers
215
views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as $n\...
- 11
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
0
votes
1
answer
738
views
Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?
Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...
- 1,236
1
vote
1
answer
510
views
The space of generalized complex structures in sense of N.Hitchin is contractible?
Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...
user21574
1
vote
2
answers
674
views
Non simply connected HyperKähler 4-manifolds without ALE metrics
In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?
- 1,236
2
votes
1
answer
213
views
What happens to small squares in Riemann mapping?
I have a square S, and I want to convert it to the unit disc D.
The Riemann mapping theorem says that I can do this with a conformal bijective map. But, any such mapping will cause some distortion.
...
- 3,549
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
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
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
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
1
vote
1
answer
218
views
Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold
Hallo,
Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...
- 339
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
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
0
votes
1
answer
339
views
Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
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
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
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
2
votes
1
answer
425
views
holomorphic extension of forms
hallo,
I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore $\...
- 29
1
vote
1
answer
397
views
Einstein metrics on the tangent bundle
Let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. Does the tangent bundle always admit an Einstein metric ?
- 213
3
votes
1
answer
345
views
extended forms from foliations [closed]
hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...
- 221
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
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
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
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
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
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