Conformal geometry and complete minimal surfaces

Kusner, Rob

doi:10.1090/s0273-0979-1987-15564-9

Cited by 61 publications

(58 citation statements)

References 7 publications

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“…When the surface is immersed, there is a more general bound (due to Li and Yau [24], sharpened in [20]…”

Section: The Minimax Sphere Eversionmentioning

confidence: 99%

“…This inspired Kusner to find an infinite family of W -critical spheres (with even-order cyclic symmetry) and real projective planes (with odd-order symmetry). These are Möbius images of complete minimal surfaces with planar ends for which there is explicit, symmetric Weierstrass data [20]. They include a W -minimizing Boy surface with 3-fold symmetry, as well as the Morin surface of 4-fold symmetry that we use as our halfway model h 0 .…”

Section: The Minimax Sphere Eversionmentioning

confidence: 99%

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The Minimax Sphere Eversion

Francis

Sullivan

Kusner

et al. 1997

Visualization and Mathematics

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We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin's classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE automatic virtual environment at Supercomputing'95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking.

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“…When the surface is immersed, there is a more general bound (due to Li and Yau [24], sharpened in [20]…”

Section: The Minimax Sphere Eversionmentioning

confidence: 99%

Section: The Minimax Sphere Eversionmentioning

confidence: 99%

The Minimax Sphere Eversion

Francis

Sullivan

Kusner

et al. 1997

Visualization and Mathematics

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“…The first statement of this theorem was also proved independently by R. Kusner [11]. The second statement follows easily from the first using Schoen's characterization of the catenoid [14].…”

Section: Theorem 6 a Complete Connected Minimal Surface M Of Finite mentioning

confidence: 69%

Properties of properly embedded minimal surfaces of finite topology

Hoffman¹,

Meeks²

1987

Bull. Amer. Math. Soc.

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Until the recent discovery of a sequence of properly embedded minimal surfaces with finite topology (Hoffman [4, 5]; Hoffman and Meeks [6, 7]), the only known examples were the plane, the catenoid and the helicoid. The existence of these new examples, which we will call Mk, k > 1, and others we have found (Callahan, Hoffman and Meeks [1]) makes it natural to ask qualitative questions about their behavior.It is a fundamental fact, due to Osserman [12], that if a complete minimal surface has finite total curvature, then it is, conformally, a closed Riemann surface punctured in a finite number of points; finite total curvature implies finite topology. The minimal surface Mk has total curvature equal to -47T(A; + 2). It is conformally a closed surface of genus k punctured in three points.A natural question to ask is whether or not each of the surfaces Mk lies in a one-parameter family of complete embedded minimal surfaces of finite total curvature. It is known that the plane and the catenoid are the unique embedded examples of finite total curvature with their respective topologies. In particular they cannot be perturbed through embedded examples. However In the case of genus k = 1, the surface Mi is conformally the square torus C/Z x Z, punctured in the three half-lattice points. It can be deformed through a family of embedded minimal surfaces which are, conformally, rectangular tori punctured in the three half-lattice points. In general, the surfaces Mk have two catenoid-type ends and one flat end. (By an end of a complete minimal surface of finite total curvature, we mean the image in R 3 of a neighborhood of a puncture point. An embedded end is a "catenoid-type end" if it converges at infinity to an end of the catenoid. It is called a "flat end" if it converges at infinity to a plane. For embedded ends on complete minimal surfaces of finite total curvature, these are the only possibilities.) They contain k + 1 straight lines which meet at a common point P and diverge into the flat end. The perturbations have three catenoid-type ends and contain no lines. The symmetry group of Mk is the dihedral group D(2k + 2), generated

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“…Nayatani (private communication) has pointed out to us that an example written down by Kusner [44], Rosenberg-Toubiana [64,65], R. Bryant [6,7] with 4 flat ends, genus 0, and Gauss map of degree d = 3, gives strict inequality. On any surface the support function u = X · N satisfies Lu = 0, where L is defined in (7.2), and because the ends are flat, u is bounded.…”

Section: Theorem 74 ([54]) the Index Of Stability Ofmentioning

confidence: 99%