All Questions
Tagged with complex-geometry elliptic-pde
15 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
459
views
A question on certain elliptic PDE
Consider the elliptic PDE
$$(CR)\;\;\;\;\;\;\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence $$(LAP)\;\;\;\;\;\;U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are ...
- 356
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
5
votes
1
answer
183
views
Can harmonic maps with immersive boundary conditions have singular points?
Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be ...
- 6,741
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: $$\text{tr}_g(\hat{g}) \leq \text{tr}_{\hat{g}} (...
user105074
3
votes
0
answers
107
views
Complex Monge-Ampere equation with degenerate right hand side
Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...
- 31
2
votes
1
answer
338
views
Existence of non-constant solutions for this equations
This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
- 601
2
votes
1
answer
160
views
The Monge- Ampère equation with a non positive right hand side
Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the ...
- 153
2
votes
0
answers
63
views
A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
- 809
2
votes
0
answers
89
views
Green’s function vector bundle laplacian
On a compact Riemann surface with a metric, there exists a Green’s function $C ln(d(x,y)^2)\leq G(x,y)\leq 0$ satisfying $u=\int u+ \int G(x,y) u(y) dy$.
Suppose $(E,h)$ is a Hermitian holomorphic ...
- 3,383
2
votes
0
answers
324
views
Siu's arguments on Calabi-Yau theorem?
In Siu's lecture note Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, he shows the $C^0$ and $C^2$ estimates of the complex Monge-Ampère equation on a Riemannian ...
- 21
1
vote
0
answers
97
views
$L^{\infty}$ estimate for bounded function on complex manifold with conic Kähler metric
Let $\overline{X}$ be a compact Kähler manifold of complex dimention $n$ with normal crossing divisors $D=\sum_{i=1}^{m}=D_{i}$. For $0<\alpha< 2$, we can construct a conic Kähler metric by ...
- 206
0
votes
0
answers
65
views
To study the elliptic PDE on complex manifold, when can we treat it as the real case?
I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying
$$\Delta_c u = f(x,u),$...
- 809
0
votes
0
answers
124
views
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