Translating graphs by mean curvature flow

Abstract

The aim of this work is studying translating graphs by mean curvature flow in R 3 . We prove non-existence of complete translating graphs over bounded domains in R 2 . Furthermore, we show that there are only three types of complete translating graphs in R 3 ; entire graphs, graphs between two vertical planes, and graphs in one side of a plane. In the last two types, graphs are asymptotic to planes next to their boundaries. We also prove stability of translating graphs and then we obtain a pointwise curvature bound for translating graphs in R 3 .

Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

“…Moreover the flow parameter is given by where ∇ P and div P are, respectively, the Riemannian connection and divergence in (P, g 0 ). This notion of translating soliton has been extensively studied in Euclidean spaces, see for instance [29], [4], [14], [38], [39], [41], [45], [48], [49] only to quote a few examples of the vast literature on the subject. Our definition is the natural setting to these special flows in Riemannian products I × P with I ⊂ R.…”

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

“…It plays an important role due to the following well-known correspondence: Σ n ⊆ R n+1 is a unit speed self-translating surface in the x n+1 -direction if and only if Σ is a minimal submanifold ofR n+1 . See for instance [Sh11] for a proof in the case n = 2 or [Pé16] for the general case.…”

The boundary behavior of domains with complete translating, minimal and CMC graphs in N2×ℝ

“…We recall some curvature estimates from Schoen [16], Minicozzi-Colding Now we derive the curvature estimate for translating graphs. The idea follows from Shariyari [17]. Let i 0 denote the injective radius in N 2 .…”

“…Some geometric properties were investigated in [1,20,10,4,19] etc. Recently Shahriyari [17] showed that if Σ is a complete translating graph Date: September 24, 2018. 2010 Mathematics Subject Classification.…”