Embedded Surfaces of Arbitrary Genus Minimizing the Willmore Energy Under Isoperimetric Constraint

For a smooth closed embedded planar curve, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus greater than 1 having the given curve as boundary, without any prescription on the conormal. By general lower bound estimates, in case the curve is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and we calculate the infimum. Then we study the case in which the genus is 1 and the competitors are restricted to a suitable class of varifolds including embedded surfaces, and we prove that the non-existence of minimizers implies a lower bound on the infimum; therefore we use such criterion in order to explicitly ﬁnd an inﬁnite family of curves for which such problem does have minimizers in the corresponding class of varifolds.

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Embedded Surfaces with Ergodic Geodesic Flows

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497001199 ◽

1997 ◽

Vol 07 (07) ◽

pp. 1509-1527 ◽

Cited By ~ 8

Author(s):

Keith Burns ◽

Victor J. Donnay

Keyword(s):

Geodesic Flow ◽

Geodesic Flows ◽

Smooth Surfaces ◽

Arbitrary Genus ◽

Embedded Surfaces

Following ideas of Osserman, Ballmann and Katok, we construct smooth surfaces with ergodic, and indeed Bernoulli, geodesic flow that are isometrically embedded in R3. These surfaces can have arbitrary genus and can be made analytic.

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Mazur’s rational torsion result for pointless genus one curves: examples

Research in Number Theory ◽

10.1007/s40993-020-00231-z ◽

2021 ◽

Vol 7 (1) ◽

Author(s):

Arjan Dwarshuis ◽

Majken Roelfszema ◽

Jaap Top

Keyword(s):

Algebraic Geometry ◽

Elliptic Curves ◽

Rational Point ◽

Torsion Points ◽

Arbitrary Genus ◽

Mazur’S Theorem ◽

Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

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Stability of stationary points for one-dimensional Willmore energy with spatially heterogeneous term

Physica D Nonlinear Phenomena ◽

10.1016/j.physd.2020.132812 ◽

2020 ◽

pp. 132812

Author(s):

Masaaki Uesaka ◽

Ken-Ichi Nakamura ◽

Keiichi Ueda ◽

Masaharu Nagayama

Keyword(s):

Stationary Points ◽

One Dimensional ◽

Willmore Energy

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Isothermic constrained Willmore tori in 3-space

Annals of Global Analysis and Geometry ◽

10.1007/s10455-021-09778-1 ◽

2021 ◽

Author(s):

Lynn Heller ◽

Sebastian Heller ◽

Cheikh Birahim Ndiaye

Keyword(s):

Conformal Class ◽

Willmore Energy

AbstractWe show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $$8\pi $$ 8 π . In particular, every constrained Willmore torus with Willmore energy below $$8\pi $$ 8 π and non-rectangular conformal class is non-degenerated.

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