All Questions
Tagged with complex-geometry elliptic-pde
7 questions with no upvoted or accepted answers
3
votes
0
answers
107
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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
0
answers
63
views
A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
- 809
2
votes
0
answers
89
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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
2
votes
0
answers
324
views
Siu's arguments on Calabi-Yau theorem?
In Siu's lecture note Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, he shows the $C^0$ and $C^2$ estimates of the complex Monge-Ampère equation on a Riemannian ...
- 21
1
vote
0
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97
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$L^{\infty}$ estimate for bounded function on complex manifold with conic Kähler metric
Let $\overline{X}$ be a compact Kähler manifold of complex dimention $n$ with normal crossing divisors $D=\sum_{i=1}^{m}=D_{i}$. For $0<\alpha< 2$, we can construct a conic Kähler metric by ...
- 206
0
votes
0
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65
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To study the elliptic PDE on complex manifold, when can we treat it as the real case?
I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying
$$\Delta_c u = f(x,u),$...
- 809
0
votes
0
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124
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integral of the laplacian to some power
I want to know the space of functions where the following quantity is uniformly bounded from above
$$\int_{K} (\Delta u)^j d\lambda< C,$$
where K is a compact and j is an integer number greater ...
- 35