Translating surfaces under flows by sub-affine-critical powers of Gauss curvature

Abstract

We construct and classify translating surfaces under flows by sub-affine-critical powers of the Gauss curvature. This result reveals all translating surfaces which model Type II singularities of closed solutions to the α-Gauss curvature flow in R 3 for any α > 0.To this end, we show that level sets of a translating surface under the α-Gauss curvature flow with α ∈ (0, 1 4 ) converge to a closed shrinking curve to the α 1−α -curve shortening flow after rescaling, namely the surface is asymptotic to the graph of a homogeneous polynomial. Then, we study the moduli space of translating surfaces asymptotic to the polynomial by using slowly decaying Jacobi fields.Notice that our result is a Liouville theorem for a degenerate Monge-Amère equation, detwhere α ∈ (0, 1 4 ).1 The convex hull of the image surface Σt is the inside region, when Σt is non-compact.2 The α-Gauss curvature flow with α = 1 n+2 is the affine normal flow in R n+1 . 3 The shrinkers under the α-curve shortening flow were already classified by Andrews [6] . 4 We say that a surface is complete and convex if it is the boundary of a convex body. 5 Σ is not necessarily the graph of u. For example, if ∂Ω has a line segment L then Σ has a flat side in L × R; see the result and pictures in [32]. 6 If Σ touches ∂Ω × R, it might be neither strictly convex nor smooth on ∂Ω × R; see [32]. 7 See Definition 2.3. 8 If m ≥ 3.

Asymptotics near extinction for nonlinear fast diffusion on a bounded domain

“…The lemma provides an effective way to extract the quantized behaviour of the solution prescribed by the discrete spectrum of its limit. It plays a crucial role in recent progress on classifications of ancient solutions (solutions defined for (−∞, T ]) to parabolic equations [ADS20, ABD Š19, BC19, CM19] and entire solutions to elliptic equations [CCK21] arising from geometry. In classifications of ancient flows, the lemma is applied backward in time (i.e.…”

“…This strategy goes back to the work of Allard and Almgren [AFA81], who gave kernel integrability conditions guaranteeing that minimal surfaces converge to their tangent cones sequentially and exponenentially fast. See also Section 6 of Simon [Sim85], or the recent contributions of Choi, Choi, Kim and Sun in various combinations [CS20] [CCK21].…”

Regional classification according to PM10 concentrations in the Seoul Metropolitan and Chungcheong Area, Republic of Korea

“…Finally, we also provide a quantitative variant of the above theorems, which will play a crucial role in the following paper [15] for the classification of translators. The following corollary roughly says if the eccentricity is controlled at a sufficiently high level, then it is also controlled at higher levels in a universal way.…”

“…The first preprint of current research [12], posted in 2021, includes an existence result weaker than Theorem 1.2 and a classification theorem. The preprint, however, has an error in the classification part.…”

Classification of noncollapsed translators in $\mathbb{R}^4$

“…See [Ngu09], [HMW19a] and [HMW19b] for other examples of translators, and [HIMW19b] for a survey article about translators in R 3 . See also [CCK21] for a recent classification of translators of the α-Gauss curvature flow in R 3 .…”