[Paper Review] On the Critical Points of the E_k Functionals in Kahler Geometry

Valentino Tosatti|Jun 1, 2005

Hermeneutics and Narrative Identity8 references3 citations

TL;DR

This paper proves that any Kähler metric in the anticanonical class which is a critical point of the $E_k$ functional and has nonnegative Ricci curvature must be Kähler-Einstein, providing a partial affirmative answer to a question by X.X. Chen on the $E_1$ functional. The result relies on the minimum principle applied to symmetric functions of Ricci curvature and extends prior work on $E_n$ and $E_0$ functionals.

ABSTRACT

We prove that a Kahler metric in the anticanonical class which is a critical point of the functional E_k and has nonnegative Ricci curvature, is necessarily Kahler-Einstein. This partially answers a question of X.X.Chen.

Motivation & Objective

• To determine whether critical points of the $E_k$ functionals in Kähler geometry are necessarily Kähler-Einstein under nonnegative Ricci curvature.
• To extend previous results on $E_0$ (Mabuchi energy) and $E_n$ to general $E_k$ functionals for $0 < k < n$.
• To investigate the relationship between critical points of $E_k$ and Kähler-Einstein metrics, particularly in the anticanonical class.
• To provide evidence for Chen's conjecture that $E_1$ critical points in the anticanonical class are Kähler-Einstein, under mild curvature assumptions.
• To establish a general framework using symmetric functions of Ricci curvature and the minimum principle to analyze critical metrics.

Proposed method

• Define the $E_k$ functionals using integration over paths of Kähler potentials, with dependence on Ricci curvature and volume forms.
• Express the critical point equation as $\sigma_{k+1}(\omega_\phi) - \triangle_\phi \sigma_k(\omega_\phi) = \binom{n}{k+1} \mu_k$, where $\sigma_k$ are elementary symmetric functions of Ricci eigenvalues.
• Apply the minimum principle to the normalized symmetric functions $\Sigma_k(\omega) = \sigma_k(\omega)/\binom{n}{k}$ at a minimum point of $\Sigma_k$.
• Use the classical inequality $\Sigma_k^{k+1} \geq \Sigma_{k+1}^k$ to propagate lower bounds from $\Sigma_{k+1} \geq 1$ to $\Sigma_k \geq 1$ and ultimately to $\Sigma_1 \geq 1$.
• Relate $\Sigma_1(\omega) = R/n$ to the scalar curvature and use $\Sigma_1 \geq 1$ to deduce $R \geq n$, implying $\mathrm{Ric}(\omega) = \omega$.
• Handle the $k=1$ case with $R > -n$ by showing $R^2 \geq n^2$ at a minimum, forcing $R \geq n$ and thus $\mathrm{Ric}(\omega) = \omega$.

Experimental results

Research questions

• RQ1 Is a critical point of the $E_k$ functional in the anticanonical class with nonnegative Ricci curvature necessarily Kähler-Einstein?
• RQ2 Does the $E_1$ functional's critical points coincide with Kähler-Einstein metrics under the condition $R > -n$?
• RQ3 Can the minimum principle and symmetric function inequalities be used to prove Kähler-Einsteinity for $E_k$ critical metrics?
• RQ4 How do the $E_k$ functionals relate to stability and the existence of extremal metrics in positive first Chern class geometry?
• RQ5 To what extent can the nonnegativity of Ricci curvature be weakened or removed for $E_k$ critical metrics to remain Kähler-Einstein?

Key findings

• For $0 < k < n$, any $E_k$-critical Kähler metric in the anticanonical class with nonnegative Ricci curvature is Kähler-Einstein.
• The result holds even when Ricci curvature is only nonnegative, not necessarily positive, due to the positivity of $\Sigma_{k+1}$ at the minimum point.
• For $k = 1$, the condition $R > -n$ suffices to ensure $E_1$-critical metrics are Kähler-Einstein, extending the result beyond positive Ricci curvature.
• The critical equation for $k=1$ reduces to $R^2 - |\mathrm{Ric}|^2 - 2\triangle R = n(n-1)$, and the minimum principle forces $R \geq n$.
• The normalized scalar curvature $\Sigma_1(\omega) \geq 1$ implies $R \geq n$, which forces $\mathrm{Ric}(\omega) = \omega$, proving Kähler-Einsteinity.
• The proof generalizes to $k = n$, where $\triangle \sigma_n = 0$ implies $\sigma_n = 1$, and the same chain of inequalities leads to $\mathrm{Ric}(\omega) = \omega$.