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"Hypersurfaces."
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Intersections of Rotational Quadratic Hypersurfaces with a Common Focus
This paper deals with rotational quadratic hypersurfaces in an n-dimensional Euclidean space. Namely, we explore some basic properties of the intersection of two rotational quadratic hypersurfaces that have a common focus, but are not necessarily confocal. We prove that any such intersection lies in at most two hyperplanes, and we specify the maximum number of its connected components.
Journal Article
Area of Minimal Hypersurfaces in the Unit Sphere (II)
In this paper, we estimate areas of compact minimal rotational hypersurfaces in the unit sphere. A sharper lower bound and upper bound on areas of these hypersurfaces are given. As applications, we confirm two conjectures, one in Perdomo and Wei (Nonlinear Anal 125:241–250, 2015), another in Cheng et al (Asian J Math 25:183–194, 2021) and give estimations of entropies of some special self-shrinkers.
Journal Article
Parallel Hypersurfaces in 4 and Their Applications to Rotational Hypersurfaces
This study explores parallel hypersurfaces in four-dimensional Euclidean space E4, deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry.
Journal Article
Parallel Hypersurfaces in Esup.4 and Their Applications to Rotational Hypersurfaces
This study explores parallel hypersurfaces in four-dimensional Euclidean space E[sup.4] , deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry.
Journal Article
Generic mean curvature flow I; generic singularities
It has long been conjectured that starting at a generic smooth closed embedded surface in R 3 , the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.
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Totally Geodesic and Parallel Hypersurfaces of Cahen-Wallach Spacetimes
We completely classify and describe totally geodesic hypersurfaces of Cahen-Wallach spacetimes. We also describe parallel hypersurfaces and investigate their geometric properties.
Journal Article
Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M
This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: the cases |a| ≪ M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal |a| < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al. (ed.), World Scientific, Singapore, 2011, pp. 132–189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincaré], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notation so that it can be read independently of previous work.
Journal Article
Stable Capillary Hypersurfaces in a Half-Space or a Slab
We study stable immersed capillary hypersurfaces in a domain 𝓑 that is either a half-space or a slab in the Euclidean space ℝn+1. We prove that such a hypersurface Σ is rotationally symmetric in the following cases: (1) n = 2, 𝓑 is a slab and Σ has genus zero. (2) n ≥ 2, 𝓑 is a slab, the angle of contact is π/2, and each component of ∂Σ is embedded. (3) n ≥ 2, 𝓑 is a half-space, the angle of contact is < π/2, and each component of ∂Σ is embedded. Moreover, in case (2), if not a right circular cylinder, Σ has to be graphical over a domain in ∂𝓑. In case (3), Σ is a spherical cap.
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