All Questions
33 questions with no upvoted or accepted answers
21
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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
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
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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
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
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
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 ...
- 1,035
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
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
0
answers
336
views
Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
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
2
votes
0
answers
211
views
When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
- 293
2
votes
0
answers
231
views
Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is ...
- 1,379
2
votes
0
answers
203
views
Yau proof of $K_X>0$ using a non-smooth metric which restricts to a metric of negative holomorphic sectional curvature on all curves
In this lecture of Yau's on the Existence of complete Kähler-Einstein metrics with negative scalar curvature he mentions the following, I quote:
Negative holomorphic sectional curvature is a rather ...
- 1,379
2
votes
0
answers
119
views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
- 21
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)>...
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
1
vote
0
answers
100
views
Curvature and Hermitian-Einstein conditions
The following is from a set of lecture notes I'm following and I have had some difficulties understanding it.
Let us discuss a few equivalent formulations of the Hermite-Einstein condition ($\Lambda_\...
- 111
1
vote
0
answers
112
views
Mean curvature as a contraction
I'm going over some of Kobayashi's work on complex vector bundles and trying to state some of the notions in a more familiar language to me.
The set up is the following. We have a hermitian vector ...
- 103
1
vote
0
answers
62
views
Expression of the Riemannian metric on the Siegel domain?
I'm looking for proof that, for the complex Siegel domain in $\mathbb C^{n}$ defined by:
$$\mathcal H_{n} = \{ z=(z_1,\dots,z_n) \in \mathbb C^{n} \mid \operatorname{Im}(z_{n}) > \sum_{j=1}^{n-1} |...
- 650
1
vote
0
answers
180
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Conceptual understanding of the definition for Hermite-Einstein metrics
I'm studying holomorphic vector bundles $(E,h)$ on Kähler manifolds that admit a Hermite-Einstein metrics. Particularly, I'm trying to find the motivation for the definition.
An hermitian structure $...
- 73
1
vote
0
answers
172
views
Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563
1
vote
0
answers
55
views
What are we to deduce from a structure theorem of this type concerning totally geodesic maps?
I apologise in advance for the vague nature of the question, but some insight would be greatly appreciated.
I'm reading a paper of Lei Ni concerning structure theorems for Kähler manifolds. Here is an ...
- 651
1
vote
0
answers
86
views
Preservation of the complex structure in warped product
Let $M=N \times_fF$ a (real) warped product submanifold of a Kähler manifold $W$.
If $M$ does not preserve the complex structure, is it possible that $N$ or $F$ preserve it? Or surely not even them?
...
- 272
1
vote
0
answers
162
views
Warped product manifold with real and complex parts
Is possible to define a warped product manifold $M=(N,g_N) \times f(F, g_F)$ where $(N, g_N)$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $(F, g_F)$ ...
- 272
1
vote
0
answers
497
views
(Real) holomorphic vector fields on compact Kähler manifolds
I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...
- 11
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
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
0
votes
0
answers
161
views
covariant derivative of a function
Let $f$ be a smooth function such on a compact kahler manifold $(M, w)$, and the component of $w$ is denoted by $g_{ij}$, assume there is a constant $s$ such that $sf = -g^{ij}\sqrt{-1}\partial_{j}\...
- 101
0
votes
0
answers
135
views
Real-Complex warped product
I have a warped product $M=N_1\times_f N_2$ where $N_1$ and $N_2$ are Riemannian manifolds.
The dimension of $N_2$ is $2n$ (for n integer) and $N_2$ is an almost Hermitian manifold, i.e., is ...
- 272
0
votes
0
answers
149
views
Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold
Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
- 1