[Paper Review] On the Fill-in of Nonnegative Scalar Curvature Metrics

Yuguang Shi, Wenlong Wang+2 more|arXiv (Cornell University)|Jul 29, 2019

Geometric Analysis and Curvature Flows22 references4 citations

TL;DR

This paper investigates the existence of fill-in metrics with nonnegative scalar curvature (NNSC) for Bartnik data $(\Sigma^{n-1}, \gamma, H)$, proving that large mean curvature $H$ or large total mean curvature $\int H\,d\mu_\gamma$ rules out NNSC fill-ins when $\gamma$ is isotopic to the standard metric on $\mathbf{S}^{n-1}$, and establishes sufficient conditions for positive scalar curvature (PSC) fill-ins via the $\theta$-invariant and gluing techniques in dimensions $3 \leq n \leq 7$. The results partially confirm a conjecture by Gromov on mean curvature bounds in scalar curvature geometry.

ABSTRACT

In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $(Σ,γ,H)$. We prove that given a metric $γ$ on $\mathbf{S}^{n-1}$ ($3\leq n\leq 7$), $(\mathbf{S}^{n-1},γ,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough (Theorem ef{Thm: no fillin nonnegative scalar 2}). Moreover, we prove that if $γ$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $\mathbf{S}^{n-1}$, then the much weaker condition that the total mean curvature $\int_{\mathbf S^{n-1}}H\,\mathrm dμ_γ$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P.\,23 in \cite{Gromov4}). In the second part of this paper, we investigate the $θ$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

Motivation & Objective

• To determine conditions under which Bartnik data $(\Sigma^{n-1}, \gamma, H)$ admits a fill-in metric with nonnegative scalar curvature.
• To investigate the role of mean curvature $H$ and total mean curvature $\int H\,d\mu_\gamma$ in obstructing NNSC fill-ins.
• To provide a partial affirmative answer to Gromov's conjecture on mean curvature bounds in scalar curvature geometry.
• To establish sufficient conditions for the existence of positive scalar curvature (PSC) fill-ins using the $\theta$-invariant and geometric gluing techniques.
• To extend results from 3D to higher dimensions ($3 \leq n \leq 7$) for scalar curvature fill-in problems.

Proposed method

• Uses a Schwarzschild neck construction to build initial PSC metrics with prescribed boundary mean curvature.
• Applies a gluing procedure along a hypersurface to combine two PSC metrics with matching boundary data, preserving positive scalar curvature.
• Employs a $\theta$-invariant to characterize the existence of PSC fill-ins for Bartnik data.
• Utilizes isotopy classes of positive scalar curvature metrics on $\mathbf{S}^{n-1}$ to relate different boundary metrics.
• Applies a rescaling argument to transfer PSC fill-ins across metrics scaled by positive constants.
• Relies on the existence of a $\delta$-collar neighborhood with a warped product structure to facilitate smooth gluing.

Experimental results

Research questions

• RQ1 Under what conditions does a triple $(\Sigma^{n-1}, \gamma, H)$ with $\gamma$ isotopic to the standard metric on $\mathbf{S}^{n-1}$ admit a fill-in with nonnegative scalar curvature?
• RQ2 Can a large pointwise mean curvature $H$ rule out the existence of NNSC fill-ins for $\Sigma^{n-1} = \mathbf{S}^{n-1}$?
• RQ3 Does a large total mean curvature $\int_{\mathbf{S}^{n-1}} H\,d\mu_\gamma$ rule out NNSC fill-ins when $\gamma$ is PSC and isotopic to the standard metric?
• RQ4 What conditions on the $\theta$-invariant ensure the existence of a positive scalar curvature fill-in for Bartnik data?
• RQ5 Can PSC fill-ins be constructed for metrics $\gamma_0$ isotopic to $\gamma_1$ when $\gamma_1$ admits a PSC fill-in with negative mean curvature?

Key findings

• If $H$ is sufficiently large on $\mathbf{S}^{n-1}$, then $(\mathbf{S}^{n-1}, \gamma, H)$ admits no fill-in with nonnegative scalar curvature, provided $3 \leq n \leq 7$.
• If $\gamma$ is a positive scalar curvature metric on $\mathbf{S}^{n-1}$ isotopic to the standard metric, then a large total mean curvature $\int_{\mathbf{S}^{n-1}} H\,d\mu_\gamma$ rules out NNSC fill-ins, providing a partial confirmation of Gromov's conjecture.
• For $\gamma_0$ and $\gamma_1$ isotopic in the space of PSC metrics on $\mathbf{S}^{n-1}$, if $(\Sigma^{n-1}, \gamma_1, -H_1)$ admits a PSC fill-in for $H_1 < \left(\frac{n-1}{n-2}\min R_{\gamma_1}\right)^{1/2}$, then $(\Sigma^{n-1}, \gamma_0, H)$ admits a PSC fill-in for all $H < \left(\frac{n-1}{n-2}\min R_{\gamma_0}\right)^{1/2}$.
• The existence of a PSC fill-in for $(\Sigma^{n-1}, \gamma, -H)$ with positive constant $H$ implies the existence of a PSC fill-in for $(\Sigma^{n-1}, \gamma, 0)$, under the condition $R_\gamma > \frac{n-2}{n-1}H^2$.
• A $\delta$-collar neighborhood with a warped product metric $g = dr^2 + q(r)\gamma_1$ is used to construct smooth gluing of PSC metrics across boundary slices.
• The $\theta$-invariant provides a sufficient condition for the existence of PSC fill-ins, generalizing results from 3D to higher dimensions.