0
votes
0answers
79 views

Commuting Derivative and Convolution type integral

Suppose $\Gamma$ is a smooth curve, $f$ and its derivative belong to some $L^p(\mathbb{C})$(i.e $f\in W^{1,p}$) and kernel $K(|z-y|)\in\mathbb{C}\times\mathbb{C}$ has only singularity on ...
1
vote
1answer
231 views

A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...
3
votes
1answer
161 views

Does $\partial\overline{\partial}f=0$ imply $f\equiv c$ for particular kind of $f$?

Hi! Let $f\in C^{2,\alpha}\left( \mathbb{C}^{m}\setminus \overline{B_{R}},\mathbb{R} \right)$ with $m\geq 2$, $R>0$ and s.t. $f$ has an expansion of type $$f=1+\mathcal{O}\left( \frac{1}{|z|} ...
8
votes
1answer
685 views

Question about an estimate in Hörmander's proof of Cartan's Theorem B

I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...
1
vote
0answers
236 views

Strong minimum principle for maximal plurisubharmonic functions

Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
3
votes
3answers
638 views

Monge Ampere equations

I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
29
votes
5answers
3k views

Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...
6
votes
0answers
356 views

A non-elliptic PDE

I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X : $\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial ...
6
votes
3answers
630 views

Newlander-Nirenberg for surfaces

Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest possible case, namely ...