Immortal Curve Shortening Flow on Analytic Surfaces
We show that on any closed orientable real-analytic Riemannian surface, any smoothly immersed closed immortal curve-shortening flow converges to a unique non-constant closed geodesic.
math.DG↗
arXiv subjects
Aprameya Girish Hebbar
Publications and source records attributed to Aprameya Girish Hebbar.
Asymptotic Geometry of Four-Dimensional Steady Solitons
In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak $κ$-solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional steady solitons constructed by Lai.
math.DG↗
$κ$-solutions with the round cylinder as an asymptotic shrinker
We show that $κ$-solutions to the Ricci flow in dimensions $n\geq 4$ whose asymptotic shrinking Ricci soliton is the round cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$ must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman's ancient solution.
math.DG↗