Entire downward solitons to the scalar curvature flow in Minkowski space

Abstract

Existence and uniqueness in Minkowski space of entire downward translating solitons with prescribed values at infinity for a scalar curvature flow equation. The radial case translates into an ordinary differential equation and the general case into a fully non-linear elliptic PDE on R n .

The prescribed curvature problem for entire hypersurfaces in Minkowski space

“…If this claim is true, following the same argument as Proposition A.2 in [6], we can prove Lemma 20. We will prove this claim below.…”

“…Here, we have used lim r→∞ z r = 0, which is a direct consequence of Proposition 18. Next Lemma is a generalization of Proposition A.2 in [6].…”

“…We also denote C = 1 − 1 C 2 as in Theorem 6. The following theorem is a generalization of Theorem 1 in [6].…”

“…By a small modification of the proof of Proposition 2.1 in [6], we obtain Proposition 18. Under the hypotheses of Theorem 17, there exists a solution y of (6.3), which is defined on [0, +∞) and smooth on (0, +∞), such that…”

“…Since the proof is a small modification of the proof of Proposition 2.1 in [6], we skip it here. Now, let's study the asymptotic behavior of y.…”