[Paper Review] Semilinear equations, the $γ_k$ function, and generalized Gauduchon metrics

Jixiang Fu, Zhizhang Wang+1 more|arXiv (Cornell University)|Oct 11, 2010

Geometry and complex manifolds16 references3 citations

TL;DR

This paper introduces generalized Gauduchon metrics via the $\gamma_k$ function, solving a semilinear PDE to establish the existence of $k$-th Gauduchon metrics on compact complex manifolds. It proves that for any hermitian metric, there exists a unique conformal factor $e^v$ such that $\partial\bar{\partial}(e^v\omega^k)\wedge\omega^{n-k-1} = \gamma_k e^v \omega^n$, with $\gamma_k$ a constant obstruction, extending Gauduchon's classical result to non-Kähler settings.

ABSTRACT

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $γ_k$ functions on the space of its hermitian metrics.

Motivation & Objective

• To generalize Gauduchon metrics beyond the $k=n-1$ case by introducing $k$-th Gauduchon metrics satisfying $\partial\bar{\partial}(\omega^k)\wedge\omega^{n-k-1}=0$.
• To address the existence problem of such metrics in the conformal class of a given hermitian metric by solving a semilinear PDE involving the $\gamma_k$ function.
• To define and analyze the $\gamma_k$ function as a conformal invariant that measures obstruction to the existence of $k$-th Gauduchon metrics.
• To prove that for any compact hermitian manifold, there exists a unique $\gamma_k$ and a solution $v$ to the conformal equation $\partial\bar{\partial}(e^v\omega^k)\wedge\omega^{n-k-1} = \gamma_k e^v \omega^n$, generalizing Gauduchon's result.
• To demonstrate the existence of 1-Gauduchon metrics on specific non-Kähler manifolds, including $X_\Lambda = \#_k(S^3\times S^3)$ and $S^5\times S^1$, and show that no pluriclosed or balanced metrics exist on $S^5\times S^1$.

Proposed method

• Introduces the $\gamma_k$ function as a constant associated with a hermitian metric $\omega$, defined via the conformal equation $\partial\bar{\partial}(e^v\omega^k)\wedge\omega^{n-k-1} = \gamma_k e^v \omega^n$.
• Reduces the problem to solving a semilinear elliptic PDE of the form $\Delta v + \psi(|\nabla v|^2) + \langle B, dv \rangle = f + c$, where $\psi(t) = t$ recovers the $\gamma_k$ case.
• Applies a priori estimates and existence theory for semilinear equations on compact manifolds, leveraging the limit condition $\liminf_{t\to\infty} \psi(t)/t^\mu \geq \nu > 0$ with $\mu > 1/2$.
• Uses the divergence theorem and integration by parts to derive the compatibility condition for the solvability of the PDE, ensuring the constant $c$ is uniquely determined.
• Constructs explicit hermitian metrics $\omega_0$ on $X_\Lambda = \#_k(S^3\times S^3)$ and $S^5\times S^1$ to compute $\gamma_1$ and verify its sign.
• Applies topological obstructions (e.g., vanishing cohomology, existence of homologous-to-zero hypersurfaces) to rule out balanced or pluriclosed metrics on $S^5\times S^1$.

Experimental results

Research questions

• RQ1 Does a $k$-th Gauduchon metric exist for $1 \leq k \leq n-2$ on a compact non-Kähler complex manifold?
• RQ2 Can the existence of such metrics be guaranteed in the conformal class of a given hermitian metric via a solution to a semilinear PDE?
• RQ3 What is the role of the $\gamma_k$ function as a conformal invariant and obstruction to the existence of $k$-th Gauduchon metrics?
• RQ4 Can explicit examples of manifolds admit 1-Gauduchon metrics while lacking pluriclosed or balanced metrics?
• RQ5 What topological or geometric conditions prevent the existence of pluriclosed or balanced metrics on certain compact complex manifolds?

Key findings

• For any compact $n$-dimensional hermitian manifold and any $1 \leq k \leq n-1$, there exists a unique constant $\gamma_k$ and a smooth function $v$ (unique up to a constant) such that $\partial\bar{\partial}(e^v\omega^k)\wedge\omega^{n-k-1} = \gamma_k e^v \omega^n$, generalizing Gauduchon's result.
• When $k = n-1$, $\gamma_{n-1} = 0$, recovering Gauduchon's original theorem; when $\omega$ is Kähler, $\gamma_k = 0$ for all $k$.
• On $X_\Lambda = \#_k(S^3\times S^3)$, $\gamma_1 = -1$, so $-1 \in \Xi_1(X_\Lambda)$, and $\Xi_1(X_\Lambda) = \{-1, 0, 1\}$, proving existence of 1-Gauduchon metrics.
• On $S^5\times S^1$, $\gamma_1 < 0$, so $-1 \in \Xi_1(S^5\times S^1)$, and $0 \in \Xi_1(S^5\times S^1)$, confirming existence of 1-Gauduchon metrics.
• There is no pluriclosed metric on $S^5\times S^1$, as shown by a contradiction using $\int \partial\bar{\partial}\omega \wedge \omega_0 < 0$ and positivity of $\omega \wedge \pi^*\omega_{FS}^2$, and no balanced metric exists due to the presence of a homologous-to-zero complex hypersurface.
• The $\gamma_k$ function is a conformal invariant: $\gamma_k$ depends only on the conformal class of $\omega$, and $\gamma_k = 0$ if and only if a $k$-th Gauduchon metric exists in that class.