# On the Convergence of Non-Integer Linear Hopf Flow

@inproceedings{Guilfoyle2022OnTC, title={On the Convergence of Non-Integer Linear Hopf Flow}, author={Brendan Guilfoyle and M. A. Robson}, year={2022} }

. The evolution of a rotationally symmetric surface by a linear com- bination of its radii of curvature equation is considered. It is known that if the coeﬃcients form certain integer ratios the ﬂow is smooth and can be integrated explicitly. In this paper the non-integer case is considered for certain values of the coeﬃcients and with mild analytic restrictions on the initial surface. We prove that if the focal points at the north and south poles on the initial surface coincide, the ﬂow…

## References

SHOWING 1-10 OF 25 REFERENCES

Evolving to non-round Weingarten spheres: integer linear Hopf flows

- MathematicsPartial Differential Equations and Applications
- 2021

In the 1950's Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper we…

A quasiconformal Hopf soap bubble theorem

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

We show that any compact surface of genus zero in $${\mathbb {R}}^3$$
R
3
that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old…

Inverse mean curvature evolution of entire graphs

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

We study the evolution of strictly mean-convex entire graphs over $R^n$ by Inverse Mean Curvature flow. First we establish the global existence of starshaped entire graphs with superlinear growth at…

PARABOLIC CLASSICAL CURVATURE FLOWS

- MathematicsJournal of the Australian Mathematical Society
- 2017

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of…

Rotationally symmetric translating solutions to extrinsic geometric flows

- Mathematics
- 2021

Analogous to the bowl soliton of mean curvature flow, we construct rotationally symmetric translating solutions to a very large class of extrinsic curvature flows, namely those whose speeds are…

Asymptotic behavior of flows by powers of the Gaussian curvature

- Mathematics
- 2016

We consider a one-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $- K^\alpha \nu$, where $\nu$ denotes the outward-pointing unit normal vector and $\alpha…

Pinching estimates and motion of hypersurfaces by curvature functions

- Mathematics
- 2004

Abstract Second derivative pinching estimates are proved for a class of parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric…

A characterization of Weingarten surfaces in hyperbolic 3-space

- Mathematics
- 2007

We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^{3})$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is…

Extrinsic Geometric Flows

- MathematicsExtrinsic Geometry of Foliations
- 2021

In the chapter we study the metrics g t satisfying the Extrinsic Geometric Flow equation (see Sect. 3.2 Sections 3.4 and 3.5 collect results about existence and uniqueness of solutions (Theorems 3.1…