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

Guilfoyle, Brendan; Klingenberg, Wilhelm

doi:10.1007/s42985-021-00128-1

Cited by 6 publications

(7 citation statements)

References 25 publications

(31 reference statements)

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“…1) where a, b, c ∈ R, X and n are the position and normal vectors of the evolving surface, and r 1 , r 2 are the radii of curvature of the surface. In this paper we investigate the time evolution of a strictly convex topological 2-sphere with an axis of rotational symmetry under the above linear Hopf flow [6]. The flow is parabolic when b > 0.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In [6] the linear Hopf flow was completely solved when the flow slope −a/b takes one of the values (1.2) − a/b = 2n + 3, with n ∈ N. It was proven that the fate of an initial smooth sphere is entirely determined by the local geometry of its isolated umbilic points, in particular the order of vanishing of the difference between the radii of curvature at the poles:…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…The case at θ = π follows by reflection. For this reason the work undertaken in [6] investigated the linear Hopf flow with −a/b restricted to an odd integer ≥ 3, on smooth initial surfaces in W . The central result is then: Theorem 3.2.…”

Section: Rotationally Symmetric Surfacesmentioning

confidence: 99%

“…The central result is then: Theorem 3.2. [6] Let S 0 ∈ W be a smooth initial surface with equal umbilic slopes µ = µ 0 = µ π . The linear Hopf flow (1.1) and (1.2) behaves in the following manner:…”

Section: Rotationally Symmetric Surfacesmentioning

confidence: 99%

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On the Convergence of Non-Integer Linear Hopf Flow

Guilfoyle¹,

Morgan²

2022

Preprint

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The evolution of a rotationally symmetric surface by a linear combination 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 converges to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the fall-off of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow.The proof uses the spectral theory of singular Sturm-Liouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Introduction and Resultsmentioning

confidence: 99%

Section: Rotationally Symmetric Surfacesmentioning

confidence: 99%

Section: Rotationally Symmetric Surfacesmentioning

confidence: 99%

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On the Convergence of Non-Integer Linear Hopf Flow

Guilfoyle¹,

Morgan²

2022

Preprint

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“…Umbilic points have many subtle properties -they can control the evolution of surfaces under curvature flows [7], while Hamburger's index bound does not hold for even small perturbations of the ambient Euclidean metric [4].…”

Section: Introductionmentioning

confidence: 99%

Roots of Polynomials and Umbilics of Surfaces

Guilfoyle¹,

Klingenberg²

2022

Preprint

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For certain polynomials we relate the number of roots inside the unit circle of with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space.In particular, for N > 0 we prove that for a certain (N +2)-real dimensional family of complex polynomials of degree N , the number of roots inside the unit circle is less than or equal to 1 + N/2. This bound is established as follows.From the polynomial we construct a convex real analytic surface containing an isolated umbilic point, such that the index of the umbilic point is determined by the number of roots of the polynomial that lie inside the unit circle. The bound on the number of roots then follows from Hamburger's bound on the index of an isolated umbilic point on a convex real analytic surface.

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Roots of Polynomials and Umbilics of Surfaces

Guilfoyle,

Klingenberg

2023

Results Math

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

Cited by 6 publications

References 25 publications

On the Convergence of Non-Integer Linear Hopf Flow

On the Convergence of Non-Integer Linear Hopf Flow

Roots of Polynomials and Umbilics of Surfaces

Roots of Polynomials and Umbilics of Surfaces

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