All Questions
7 questions
7
votes
1
answer
472
views
Elliptic operator on compact Hermitian manifold
Let $X^n$ be a compact complex manifold, and $\omega$ be a Hermitian metric on $X$.
Define an operator $P:=i\Lambda_\omega \bar{\partial} \partial$ on the space of the smooth function $C^\infty(X, \...
- 167
5
votes
2
answers
303
views
Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)
There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs,
where $P_k$ is given by
\begin{equation}
\begin{aligned}
P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...
- 51
4
votes
1
answer
3k
views
Laplace spectrum of the $2$-Sphere [closed]
The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$ acting on functions. The eigenvalues of this ...
- 643
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
3
votes
0
answers
107
views
Complex Monge-Ampere equation with degenerate right hand side
Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...
- 31
2
votes
1
answer
338
views
Existence of non-constant solutions for this equations
This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
- 601
2
votes
0
answers
89
views
Green’s function vector bundle laplacian
On a compact Riemann surface with a metric, there exists a Green’s function $C ln(d(x,y)^2)\leq G(x,y)\leq 0$ satisfying $u=\int u+ \int G(x,y) u(y) dy$.
Suppose $(E,h)$ is a Hermitian holomorphic ...
- 3,383