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Application of spaces of subspheres to conformal invariants of curves and canal surfaces

Rémi Langevin, Jun O'Hara, Shigehiro Sakata (2013)

Annales Polonici Mathematici

We review some techniques from the Möbius geometry of curves and surfaces in the 3-sphere, consider canal surfaces using their characteristic circles, and express the conformal curvature, and conformal torsion, of a vertex-free space curve in terms of its corresponding curve of osculating circles, and osculating spheres, respectively. We accomplish all of this strictly within the framework of Möbius geometry, and compare our results with the literature. Finally, we show how our formulation allows...

Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group

Yen-Chang Huang (2016)

Analysis and Geometry in Metric Spaces

By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H1 is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also define the kinematic density for PSH(1) and show that the measure of all segments with length l intersecting a convex domain D ⊂ H1 can be represented by the p-area of the boundary...

Applications of Quaternionic Holomorphic Geometry to minimal surfaces

K. Leschke, K. Moriya (2016)

Complex Manifolds

In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface....

Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian

Kei Funano (2016)

Analysis and Geometry in Metric Spaces

We apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.

Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

Neil Seshadri (2009)

Bulletin de la Société Mathématique de France

To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J , we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M × ( - 1 , 0 ) . We consider the asymptotic expansion, in powers of a special defining function, of the volume of M × ( - 1 , 0 ) with respect to g and prove that the log term coefficient is independent of J (and any choice of contact...

Approximation of viscosity solution by morphological filters

Denis Pasquignon (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider in ℝ 2 all curvature equation ∂ u ∂ t = | D u | G ( curv ( u ) ) where G is a nondecreasing function and curv(u) is the curvature of the level line passing by x. These equations are invariant with respect to any contrast change u → g(u), with g nondecreasing. Consider the contrast invariant operator T t : u o → u ( t ) . A Matheron theorem asserts that all contrast invariant operator T can be put in a form ( T u ) ( 𝐱 ) = inf B ∈ ℬ sup 𝐲 ∈ B u ( 𝐱 + 𝐲 ) . We show the asymptotic equivalence of both formulations. More precisely, we show that all curvature equations can be obtained...

Äquiforme Analogien in der Kinematik der konjugierten Profile

Karel Drábek, Zdeněk Pírko (1984)

Aplikace matematiky

In der Arbeit wird vor allem die äquiforme Analogie zu der zweierleien Erzeugung der Hüllbahnkurve und ihre Ausnützyung zur Herleitung der H. R. Müllerschen Gleichungen gegebe. Aus der Beziehung in einer von diesen Gleichungen werden drei Typen der Aufgaben gelöst: direkte, inverse und gemischte. Weiter wird die Analogie zu dem Cauchyschen Satz über den Winkel der gemeinsamen Normale der konjugierten Profile und der Verbindungsgerade dieses Berührungspunktes mit dem 1-Pole in der gegebenen Phase...

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