[Paper Review] Translating solitons of the mean curvature flow asymptotic to hyperplanes in $\mathbb{R}^{n+1}$

Eddygledson S. Gama, Francisco Martín|arXiv (Cornell University)|Jan 1, 2018

Geometric Analysis and Curvature Flows18 references1 citations

TL;DR

This paper classifies translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder, proving they must be either hyperplanes parallel to the translating velocity or tilted grim reaper cylinders. The proof uses a moving plane argument, barrier constructions with translating catenoids, and varifold convergence to rule out non-trivial configurations, extending prior results to general dimensions under $n < 7$.

ABSTRACT

A translating soliton is a hypersurface $M$ in $\mathbb{R}^{n+1}$ such that the family $M_t= M- t \,\mathbf{e}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf{e}_{n+1}^{\perp}.$ In this paper we obtain a characterization of hyperplanes which are parallel to $\mathbf{e}_{n+1}$ and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ which are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $\mathbb{R}^3$ by the second author, Perez-Garcia, Savas-Halilaj and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

Motivation & Objective

• To classify complete, connected, properly embedded translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder.
• To extend previous classification results—previously restricted to $n=3$ and requiring bounded genus and perpendicular cylinders—to general dimensions $n < 7$.
• To establish that the only such solitons are hyperplanes parallel to the translating velocity or tilted grim reaper cylinders.
• To resolve the structure of asymptotic ends in translating solitons using geometric analysis and varifold convergence.

Proposed method

• Applies a moving plane argument (dynamical lemma) to compare translated copies of the hypersurface with itself, exploiting symmetry and maximum principle arguments.
• Uses translating catenoids $W^2_\lambda \times \mathbb{R}^{n-2}$ as barriers to control area blow-up and ensure smooth limit behavior in the varifold convergence argument.
• Employs varifold convergence techniques to analyze the limit of translated sequences, ensuring the limit is a stationary integral varifold with well-defined regular points.
• Applies Theorem 2.3 (maximum principle for translators) to rule out first-contact or distance-zero intersection cases between translated hypersurfaces and the original.
• Relies on the $C^1$-asymptotic condition to control the geometry at infinity and ensure the ends behave like half-hyperplanes.
• Uses the fact that $|A|^2 = H^2$ on the hypersurface to reduce the second fundamental form to a scalar multiple of the mean curvature, simplifying curvature analysis.

Experimental results

Research questions

• RQ1 What are the complete, properly embedded translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder?
• RQ2 Can the classification of tilted grim reaper cylinders and hyperplanes as the only such solitons be extended beyond $n=3$?
• RQ3 Does the assumption of $n < 7$ allow the use of varifold convergence and regularity theory to rule out non-trivial limit configurations?
• RQ4 How do barrier constructions with translating catenoids ensure the absence of area blow-up in the limit?
• RQ5 Can the moving plane method be adapted to non-perpendicular, tilted configurations to exclude intermediate solutions?

Key findings

• The only complete, connected, properly embedded translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder are hyperplanes parallel to $e_{n+1}$ or tilted grim reaper cylinders.
• For $n < 7$, the limit of translated sequences of the hypersurface converges to a stationary integral varifold, and regularity at the limit point implies equality of the translated and original hypersurfaces.
• The assumption of $C^1$-asymptoticity to two half-hyperplanes forces the asymptotic hyperplanes to be parallel, and ultimately forces the entire hypersurface to coincide with a single hyperplane.
• The proof relies on barrier techniques using $W^2_\lambda \times \mathbb{R}^{n-2}$ to prevent area blow-up, ensuring smooth limit behavior.
• The maximum principle (Theorem 2.3) rules out both first-contact and distance-zero intersection scenarios between translated copies and the original hypersurface.
• The conclusion that $|A|^2 = H^2$ on the hypersurface implies that the second fundamental form is totally umbilic, reducing the problem to scalar curvature analysis and enabling the use of classification results from [12].