[Paper Review] The $L_p$-Minkowski problem with super-critical exponents

Qiang Guang, Qirui Li+1 more|arXiv (Cornell University)|Mar 10, 2022

Topological and Geometric Data Analysis4 citations

TL;DR

This paper resolves the $L_p$-Minkowski problem for all super-critical exponents $p < -n-1$ by introducing a novel topological method based on the homology of the space of ellipsoids. The authors establish the existence of solutions to the Monge-Ampère equation on the sphere by proving uniform estimates and using a topological degree argument, overcoming long-standing challenges in the super-critical regime where previous methods failed due to lack of uniform bounds and non-compactness.

ABSTRACT

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. The Existence of solutions has been obtained in the sub-critical case $p&gt;-n-1$, but the problem remains widely open in the super-critical case $p

Motivation & Objective

• To resolve the long-open $L_p$-Minkowski problem in the super-critical case $p < -n-1$, where existence results were previously unknown for general positive densities.
• To overcome the lack of uniform estimates and compactness in the super-critical regime, which had hindered prior approaches based on variational or PDE methods.
• To establish the existence of solutions to the Monge-Ampère equation $\det(\nabla^2 u + uI) = f u^{p-1}$ on $\mathbb{S}^n$ for $p < -n-1$ and positive $f$.
• To develop a new topological framework based on the homology of the space of ellipsoids to handle the non-compactness and lack of normalization in blow-up sequences.

Proposed method

• Introduce a topological degree argument based on the homology of the space of ellipsoids, which provides a new mechanism to detect solutions in the super-critical case.
• Use a barrier-type function $w = A \sum h^{ii} + B \sum u_k u_{kt}$ with carefully chosen constants $A$ and $B$ to control the evolution of the Hessian and gradient terms.
• Derive a differential inequality for $w$ along the flow by differentiating the Monge-Ampère equation and applying the Ricci identity and covariant derivative identities.
• Establish uniform upper bounds on the largest eigenvalue of the Hessian $\nabla^2 u + uI$ via a maximum principle argument, using the structure of the equation and the choice of $A$ and $B$.
• Apply a blow-up analysis and topological degree theory to show that the degree is non-zero, implying existence of at least one solution.
• Use the projective invariance and symmetry properties of the equation to justify the topological setup and ensure the degree is well-defined.

Experimental results

Research questions

• RQ1 Does the $L_p$-Minkowski problem admit solutions for all $p < -n-1$ when the prescribed measure is a positive continuous function on $\mathbb{S}^n$?
• RQ2 Can a topological method based on the homology of the space of ellipsoids be used to prove existence in the super-critical $L_p$-Minkowski problem where standard PDE methods fail?
• RQ3 Is it possible to establish uniform estimates for the Monge-Ampère equation $\det(\nabla^2 u + uI) = f u^{p-1}$ in the super-critical regime $p < -n-1$?
• RQ4 How does the lack of a Kazdan-Warner type obstruction in the super-critical case affect the existence and multiplicity of solutions?

Key findings

• The $L_p$-Minkowski problem is solved for all super-critical exponents $p < -n-1$ with a positive density $f$ on $\mathbb{S}^n$, establishing existence of solutions to the Monge-Ampère equation.
• A new topological method based on the homology of the space of ellipsoids is successfully applied to prove existence, marking a breakthrough in the super-critical regime.
• Uniform estimates for the Hessian $\nabla^2 u + uI$ are established via a maximum principle argument on a carefully constructed auxiliary function $w$, overcoming the lack of uniform bounds in previous approaches.
• The solution is obtained using a topological degree argument that is non-zero due to the topological structure of the ellipsoid space, ensuring existence without requiring symmetry or discrete measures.
• The method applies to general positive $f \in C^0(\mathbb{S}^n)$, extending beyond previous results that were limited to symmetric or discrete measures.
• The result closes a major gap in the $L_p$-Minkowski problem, completing the existence theory across all $p \in \mathbb{R}$, including the super-critical case.