[Paper Review] On maximum-principle functions for flows by powers of the Gauss curvature

Martin Franzen|arXiv (Cornell University)|Dec 18, 2013

Geometric Analysis and Curvature Flows5 references3 citations

TL;DR

This paper proves that no maximum-principle function exists for flows by powers of the Gauss curvature when the exponent σ > 1. Using a detailed analysis of the evolution equations for symmetric rational functions of principal curvatures, the author shows that the necessary conditions for monotonicity—particularly the non-positivity of gradient and constant terms in the evolution—cannot be simultaneously satisfied, thereby ruling out the existence of such monotone quantities under these flows.

ABSTRACT

We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for proving convergence to a round point. Such monotone quantities exist for many normal velocities, including the Gauss curvature, some powers larger than one of the mean curvature, and some powers larger than one of the norm of the second fundamental form. In this paper, we show that no such quantity exists for any powers larger than one of the Gauss curvature.

Motivation & Objective

• To determine whether monotone quantities—specifically maximum-principle functions—exist for flows by powers σ > 1 of the Gauss curvature.
• To investigate whether the standard method of using maximum-principle functions can be applied to prove convergence to a round point under such flows.
• To analyze the evolution of symmetric rational functions of principal curvatures under the flow dX/dt = -K^σ ν, focusing on the sign of constant and gradient terms in the evolution equation.
• To establish that no such functions satisfying Schnürer’s criteria exist for σ > 1, despite their success in other curvature flow cases.

Proposed method

• Define a maximum-principle function w as a symmetric rational function w = p(λ₁,λ₂)/q(λ₁,λ₂), where p and q are homogeneous polynomials with specific sign and degree conditions.
• Use the evolution equation for w under the flow dX/dt = -K^σ ν, decomposing the time derivative into constant terms C_w(λ₁,λ₂) and gradient terms G_w(λ₁,λ₂)h_{ij;i}^2.
• Apply the maximum principle by requiring L(w) = d/dt w - F^{ij}w_{;ij} ≤ 0 at critical points, which reduces to checking non-positivity of C_w and G_w.
• Analyze the leading-order terms of C_w(ρ), G₁(ρ), and G₂(ρ) in terms of ρ = λ₁/λ₂, using asymptotic expansions for different cases of polynomial degrees.
• Consider multiple cases based on the leading coefficients of the numerator and denominator polynomials, systematically deriving contradictions when assuming non-positivity of all terms.
• Use the identity g - h = (k - 1)(2σ - 1)/(σ - 1) to test critical cases and show that the resulting G₂ term becomes positive, violating the required non-positivity.

Experimental results

Research questions

• RQ1 Do maximum-principle functions exist for flows by powers σ > 1 of the Gauss curvature?
• RQ2 Can the standard monotonicity method via maximum-principle functions be applied to prove convergence to a round point for K^σ flows with σ > 1?
• RQ3 What are the structural obstructions preventing the existence of such monotone quantities in the K^σ flow for σ > 1?
• RQ4 Why do the gradient and constant terms in the evolution of w fail to be non-positive simultaneously under K^σ flows for σ > 1?
• RQ5 Is there a fundamental difference in the behavior of K^σ flows for σ > 1 compared to other flows like H^σ or |A|^2, where such monotone quantities are known to exist?

Key findings

• No maximum-principle function exists for the flow dX/dt = -K^σ ν when σ > 1.
• The assumption that both the constant term C_w and the gradient terms G_w are non-positive for all ρ > 0 leads to a contradiction in all considered cases of polynomial degree combinations.
• In particular, assuming the critical identity g - h = (k - 1)(2σ - 1)/(σ - 1) leads to a positive leading coefficient in G₂(ρ), violating the required non-positivity.
• Even when relaxing this identity, the resulting conditions force g - h - 2(k - 1) > 0 and g - h - (k - 1) > 0, which still lead to a positive G₁(ρ) term, contradicting the monotonicity requirement.
• The contradiction arises regardless of the choice of leading coefficients, indicating a structural obstruction inherent to the K^σ flow for σ > 1.
• This result implies that the standard maximum-principle approach using monotone quantities cannot be used to prove convergence to a round point for these flows, although convergence itself remains an open question.