scholarly journals Isothermic constrained Willmore tori in 3-space

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.

scholarly journals A calculus for conformal hypersurfaces and new higher Willmore energy functionals

2020 ◽  
Vol 20 (1) ◽  
pp. 29-60 ◽  
Author(s):  
A. Rod Gover ◽  
Andrew Waldron
Keyword(s):  
Differential Operators ◽  
Yamabe Problem ◽  
Conformal Class ◽  
Energy Functions ◽  
Energy Functionals ◽  
Conformally Compact ◽  
Higher Dimensional ◽  
Invariant Differential ◽  
The Right ◽  
Willmore Energy

AbstractThe invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution’s asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20].

Conformal Willmore tori in ℝ4

2018 ◽  
Vol 2018 (742) ◽  
pp. 281-301
Author(s):  
Tobias Lamm ◽  
Reiner M. Schätzle
Keyword(s):  
Branch Point ◽  
Double Cover ◽  
Two Dimensional ◽  
Conformal Class ◽  
Dimensional Torus ◽  
Möbius Transformations ◽  
Energy Value ◽  
Double Covers ◽  
Willmore Energy ◽  
Mobius Transformations

Abstract For every two-dimensional torus {T^{2}} and every k \in \mathbb{N} , {k\geq 3} , we construct a conformal Willmore immersion f : T^{2} \to \mathbb{R}^{4} with exactly one point of density k and Willmore energy 4πk. Moreover, we show that the energy value {8\pi} cannot be attained by such an immersion. Additionally, we characterize the branched double covers T^{2} \to S^{2} \times \{ 0 \} as the only branched conformal immersions, up to Möbius transformations of {\mathbb{R}^{4}} , from a torus into {\mathbb{R}^{4}} with at least one branch point and Willmore energy {8\pi} . Using a perturbation argument in order to regularize a branched double cover, we finally show that the infimum of the Willmore energy in every conformal class of tori is less than or equal to {8\pi} .

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

2020 ◽  
pp. 132812
Author(s):  
Masaaki Uesaka ◽  
Ken-Ichi Nakamura ◽  
Keiichi Ueda ◽  
Masaharu Nagayama
Keyword(s):  
Stationary Points ◽  
One Dimensional ◽  
Willmore Energy

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

The large genus limit of the infimum of the Willmore energy

2010 ◽  
Vol 132 (1) ◽  
pp. 37-51 ◽  
Author(s):  
Ernst Kuwert ◽  
Yuxiang Li ◽  
Reiner Schätzle
Keyword(s):  
Large Genus ◽  
Willmore Energy

Infinitesimal bending of knots and energy change

2019 ◽  
Vol 28 (11) ◽  
pp. 1940009
Author(s):  
Louis H. Kauffman ◽  
Ljubica S. Velimirović ◽  
Marija S. Najdanović ◽  
Svetozar R. Rančić
Keyword(s):  
Visual Representation ◽  
Energy Change ◽  
Visualization Tool ◽  
Infinitesimal Bending ◽  
Willmore Energy

We discuss the infinitesimal bending of curves and knots in [Formula: see text]. A brief overview of the results on the infinitesimal bending of curves is outlined. Change of the Willmore energy, as well as of the Möbius energy under infinitesimal bending of knots is considered. Our visualization tool devoted to visual representation of infinitesimal bending of knots is presented.

Willmore energy for joining of carbon nanostructures

2018 ◽  
Vol 98 (16) ◽  
pp. 1511-1524 ◽  
Author(s):  
P. Sripaturad ◽  
N. A. Alshammari ◽  
N. Thamwattana ◽  
J. A. McCoy ◽  
D. Baowan
Keyword(s):  
Carbon Nanostructures ◽  
Willmore Energy

scholarly journals ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA

2020 ◽  
Vol 8 ◽  
Author(s):  
THIERRY DAUDÉ ◽  
NIKY KAMRAN ◽  
FRANÇOIS NICOLEAU
Keyword(s):  
Infinite Number ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Unique Continuation ◽  
Manifolds With Boundary ◽  
Conformal Class ◽  
Hölder Continuous ◽  
Continuation Principle ◽  
Partial Data ◽  
Calderón Problem

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

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