[Paper Review] A new formula for the energy functionals E_k and its applications

Haozhao Li|ArXiv.org|Sep 26, 2006

Geometry and complex manifolds8 references4 citations

TL;DR

This paper presents a new formula for Chen-Tian's energy functionals $E_k$, expressing them via a combination of curvature forms and Ricci potentials. The key contribution is a universal identity linking $E_k$ to the Kähler potential and Ricci curvature, which enables a new proof that all $E_k$-related holomorphic invariants equal the Futaki invariant, and establishes lower bounds for $E_k$ under curvature conditions.

ABSTRACT

We give a new formula for the energy functionals E_k defined by Chen-Tian, and discuss the relations between these functionals. We also apply our formula to give a new proof of the fact that the holomorphic invariants corresponding to the E_k functionals are equal to the Futaki invariant.

Motivation & Objective

• To derive a new, explicit formula for the energy functionals $E_k$ introduced by Chen and Tian.
• To clarify the structural relationships between the $E_k$ functionals and their behavior under curvature constraints.
• To prove that the holomorphic invariants associated with $E_k$ are all equal to the Futaki invariant.
• To establish lower bounds for $E_k$ under Ricci curvature conditions, generalizing prior results.
• To provide a new proof of the equivalence between $E_k$-invariants and the Futaki invariant using the derived formula.

Proposed method

• Define $E_{k, ho}^0( ho)$ as a functional involving the Ricci potential $h_\omega$, curvature forms $Ric_\varphi^i$, and Kähler forms $\omega_\varphi^{n-k}$.
• Introduce a correction term $J_{k,\omega}(\varphi)$ via a time-integral of the difference $\omega_{\varphi(t)}^{k+1} - \omega^{k+1}$.
• Define the full functional as $E_k = E_{k,\omega}^0 - J_{k,\omega}$, ensuring gauge invariance and consistency with known cases.
• Derive a universal identity: $\sum_{i=0}^k (-1)^i \binom{k+1}{i+1} E_i = \frac{1}{V} \int_M u (\sqrt{-1}\partial\bar{\partial}u)^k \wedge \omega_\varphi^{n-k} + \frac{1}{V} \int_M h_\omega (-\sqrt{-1}\partial\bar{\partial}h_\omega)^k \wedge \omega^{n-k}$, where $u = \log\frac{\omega_\varphi^n}{\omega^n} + \varphi - h_\omega$.
• Use the derived identity to prove lower bounds for $E_k$ under the curvature assumption $Ric_\varphi \geq -\frac{2}{k-1}\omega_\varphi$.
• Apply the formula to re-derive the equality of holomorphic invariants $\mathcal{F}_k = (k+1)\mathcal{F}_0$, confirming they all equal the Futaki invariant.

Experimental results

Research questions

• RQ1 How are the Chen-Tian energy functionals $E_k$ related to each other structurally?
• RQ2 Can a unified formula be derived that expresses $E_k$ in terms of curvature and potential data?
• RQ3 Do the holomorphic invariants $\mathcal{F}_k$ associated with $E_k$ coincide with the classical Futaki invariant?
• RQ4 Under what curvature conditions can $E_k$ be bounded from below?
• RQ5 Can the lower boundedness of $E_k$ be deduced from that of $E_0$?

Key findings

• A new universal formula is derived that expresses the linear combination $\sum_{i=0}^k (-1)^i \binom{k+1}{i+1} E_i$ in terms of the curvature form $\sqrt{-1}\partial\bar{\partial}u$ and the Ricci potential $h_\omega$.
• The formula generalizes Pali's result for $k=1$, recovering the known identity $2E_0 - E_1 = -\frac{1}{V}\int_M \sqrt{-1}\partial u\wedge\bar{\partial}u\wedge\omega_\varphi^{n-1} + c_1$.
• For $k \geq 2$, under the curvature condition $Ric_\varphi \geq -\frac{2}{k-1}\omega_\varphi$, the functional satisfies $E_k(\varphi) \geq (k+1)E_0(\varphi) + c_k$, where $c_k$ is a constant depending on $h_\omega$ and $\omega$.
• The holomorphic invariants $\mathcal{F}_k$ associated with $E_k$ are shown to satisfy $\mathcal{F}_k = (k+1)\mathcal{F}_0$, proving they all equal the Futaki invariant.
• The proof of this invariance is new and relies directly on the derived formula, offering an alternative to Liu's earlier proof.
• The paper establishes that $E_1$ is bounded from below if and only if $E_0$ is bounded from below, extending a result from earlier work to the full $E_k$ family.