Integral Inequality and Stability of the Positive Mass Theorem for Asymptotically Euclidean Kähler Manifolds

Keskeiset käsitteet

An integral inequality is derived that bounds the ADM mass of asymptotically Euclidean Kähler manifolds from below in terms of the scalar curvature and the Hessian of certain holomorphic coordinate functions. This inequality is then used to prove two stability results for the Positive Mass Theorem on Kähler manifolds.

Tiivistelmä

The paper establishes an integral inequality that provides a lower bound on the ADM mass of asymptotically Euclidean (AE) Kähler manifolds in terms of the scalar curvature and the Hessian of certain holomorphic coordinate functions. This inequality is then used to prove two stability results for the Positive Mass Theorem on Kähler manifolds.
Key highlights:

The integral inequality (Theorem A) states that the ADM mass is bounded from below by an integral involving the scalar curvature and the Hessian of the real and imaginary parts of the holomorphic coordinate functions arising from the complex coordinates at infinity.

The first stability result (Theorem B) shows that for any sequence of AE Kähler manifolds with vanishing ADM mass, there exist subsets with vanishing boundaries in the limit such that the complements converge to Euclidean space in the pointed Gromov-Hausdorff sense.

The second stability result (Theorem C) shows that under a uniform lower bound on the Ricci curvature and asymptotic flatness conditions, any sequence of AE Kähler manifolds with vanishing ADM mass converges to Euclidean space in the pointed Gromov-Hausdorff sense without the need to remove subsets.

Three new families of AE Kähler metrics on C^2 with vanishing mass in the limit are presented, for which the stability results apply.

Tilastot

The ADM mass of an asymptotically Euclidean Kähler manifold (X^(2m), g, J) is given by:
m(g) = -〈♣(c1(X)), [ω^(m-1)]〉/(2m-1)πm-1 + (m-1)!/(4(2m-1)π^m) ∫_X R_g dvolg
The scalar curvature R_Σ and second fundamental form II on the level sets Σ_s = {z_1 = s} satisfy:
1/2|∇x_1|^2(R_g - R_Σ - |II|^2 + 2〈Rm_X(x̂_1, x̂_2)x̂_2, x̂_1〉) = Ric(∇x_1, ∇x_1) + Ric(∇x_2, ∇x_2)

Lainaukset

"We prove an integral inequality and two stability results for the ADM mass on AE Kähler manifolds of all complex dimensions."
"The inequality bounds the ADM mass from below by an integral of the scalar curvature and the Hessian of certain holomorphic coordinate functions arising from the complex coordinates at infinity."
"This gives the first stability result of the Positive Mass Theorem for Kähler manifolds, or more generally, of manifolds without strong curvature or volume conditions or for a very explicit family of metrics in real dimensions greater than three."

Tärkeimmät oivallukset

Mass Inequality and Stability of the Positive Mass Theorem For K\"ahler Manifolds

by Johan Jacoby... klo arxiv.org 10-02-2024

https://arxiv.org/pdf/2311.05588.pdf

Syvällisempiä Kysymyksiä

What are the potential applications of the derived integral inequality and stability results beyond the Positive Mass Theorem?

The integral inequality and stability results derived in this work have several potential applications beyond the Positive Mass Theorem, particularly in the fields of geometric analysis and mathematical physics. One significant application is in the study of the asymptotic behavior of Kähler manifolds in general relativity. The stability results can be utilized to analyze the convergence of sequences of Kähler metrics, which may arise in the context of gravitational collapse or the formation of singularities.
Additionally, the integral inequality can serve as a tool for establishing lower bounds on the ADM mass in various geometric settings, potentially leading to new insights into the structure of Kähler manifolds and their curvature properties. This could have implications for the study of minimal surfaces and the behavior of holomorphic functions on complex manifolds, as the inequality provides a framework for understanding how scalar curvature interacts with the geometry of the manifold.
Moreover, the techniques developed in this work may be applicable to other geometric inequalities, such as those related to the Yamabe problem or the study of Ricci flow on Kähler manifolds. The methods could also be extended to investigate stability results in the context of other geometric theorems, such as the Penrose inequality or the Riemannian Penrose conjecture, thereby broadening the scope of research in geometric analysis.

How can the methods used in this work be extended to study the stability of other geometric theorems on Kähler manifolds?

The methods employed in this work can be extended to study the stability of other geometric theorems on Kähler manifolds by adapting the integral inequalities and stability arguments to different contexts. For instance, one could investigate the stability of the Kähler-Einstein condition by analyzing sequences of Kähler metrics that converge to a Kähler-Einstein metric. The integral inequalities derived in this paper could be modified to incorporate the specific curvature conditions required for Kähler-Einstein metrics.
Furthermore, the techniques used to establish the stability results for the Positive Mass Theorem can be applied to other stability questions, such as the stability of the Kähler-Ricci flow. By examining the behavior of the scalar curvature and the Ricci curvature under the flow, one could derive analogous stability results that characterize the convergence of Kähler metrics over time.
Additionally, the use of holomorphic coordinates and the complex coarea formula can be generalized to study the stability of other geometric structures, such as complex submanifolds or the behavior of holomorphic sections in the context of vector bundles over Kähler manifolds. This approach could lead to new insights into the interplay between complex geometry and the stability of various geometric properties.

Are there any physical or geometric interpretations of the additional curvature terms that appear in the Kähler case compared to the Riemannian case?

The additional curvature terms that appear in the Kähler case compared to the Riemannian case can be interpreted both geometrically and physically. Geometrically, these terms reflect the richer structure of Kähler manifolds, which possess both a Riemannian metric and a symplectic structure. The presence of holomorphic coordinates allows for the incorporation of complex geometric properties, leading to additional curvature contributions that do not arise in purely Riemannian settings.
From a physical perspective, these additional curvature terms can be associated with the behavior of gravitational fields in the context of general relativity. In Kähler manifolds, the interplay between the scalar curvature and the Ricci curvature can be seen as a manifestation of how the geometry of the manifold influences the distribution of mass and energy. The terms involving the second fundamental form and the Riemann curvature tensor can be interpreted as capturing the effects of gravitational interactions in a more complex geometric framework.
Moreover, these curvature terms may have implications for the stability of certain physical configurations, such as black holes or other gravitational systems modeled by Kähler metrics. Understanding how these additional terms affect the ADM mass and the stability of the manifold can provide insights into the nature of spacetime in the presence of complex structures, potentially leading to new predictions in theoretical physics.