Article
Mathematics, Applied
Tatsuya Miura
Summary: In this paper, geometric inequalities for closed surfaces in Euclidean three-space are proven, including optimal scaling laws for convex surfaces and the verification of Topping's conjecture in a class of simply-connected axisymmetric surfaces. This provides evidence that optimal shapes are necessarily straight even without convexity.
SELECTA MATHEMATICA-NEW SERIES
(2021)
Article
Mathematics
Guodong Wei
Summary: In this study, we investigate the minimizers of curvature functionals subject to an area constraint in asymptotically flat manifolds and prove the existence of such minimizers under certain conditions. Our findings may be of interest in General Relativity, as the proofs rely on various established theorems in the field.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics
Thomas Koerber
Summary: This paper investigates the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold close to Schwarzschild. It proves that the leaves of spheres in this region are stable under small area preserving perturbations, making them strict local area-preserving maximizers of the Hawking mass with respect to the W-2, W-2-topology.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Klaus Deckelnick, Marco Doemeland, Hans-Christoph Grunau
Summary: This article focuses on a special version of the Helfrich functional, discussing the existence of solutions to a Dirichlet boundary value problem for Helfrich surfaces of revolution under different boundary conditions. It is found that in certain cases, the minimizers of the Helfrich functional are connected to the global minimizers like the catenoid.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Nural Yuksel, Burcin Saltik
Summary: If the arc length and intrinsic curvature of a curve or surface are kept constant, it is considered to be inextensible. Inextensible curve and surface flows are characterized by the absence of motion-induced strain energy. This paper studies inextensible tangential, normal, and binormal ruled surfaces generated by a curve with constant torsion, known as a Salkowski curve. It investigates whether these surfaces are minimal or developable and proves theorems related to inextensible ruled surfaces in three-dimensional Euclidean space.
Article
Mathematics
Jin Wang, Zhengyuan Shi
Summary: The multi-reconstruction algorithm proposed in this study, based on a modified vector-valued Allen-Cahn equation, is able to reconstruct multi-component surfaces without overlapping or self-intersections, producing smooth surfaces and preserving the original data effectively. The algorithm involves one linear equation and two nonlinear equations, with the linear equation discretized using implicit methods and solved using fast Fourier transform. The ability to apply the algorithm directly to graphics processing units allows for faster implementation compared to traditional central processing units.
Article
Mathematics
Simon Brendle, Panagiota Daskalopoulos, Natasa Sesum
Summary: This paper studies the classification of kappa-noncollapsed ancient solutions to three-dimensional Ricci flow on S-3, and proves that such solutions are either isometric to a family of shrinking round spheres or the Type II ancient solution constructed by Perelman.
INVENTIONES MATHEMATICAE
(2021)
Article
Mathematics, Applied
Ruben Jakob
Summary: In this article, the author investigates the behavior of flow lines of the classical Willmore flow. It is proven that under certain conditions, these flow lines converge to smooth Willmore-Hopf tori and conformally transformed Clifford tori. The proof utilizes the equivariance of the Hopf fibration, the Lojasiewicz-Simon gradient inequality, and the classification and description of solutions of the Euler-Lagrange equation of the elastic energy functional.
JOURNAL OF EVOLUTION EQUATIONS
(2023)
Article
Mathematics, Applied
Junfu Yao
Summary: This short note presents a proof of the uniqueness result for small entropy self-expanders asymptotic to a fixed cone. The proof is a direct consequence of the mountain-pass theorem and the integer degree argument proved by J. Bernstein and L. Wang.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Tang-Kai Lee
Summary: This article proves that any n-dimensional closed mean convex lambda-hypersurface is convex if lambda <= 0, generalizing Guang's work on 2-dimensional strictly mean convex lambda-hypersurfaces. As a corollary, a gap theorem for closed lambda-hypersurfaces with lambda <= 0 is obtained.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Paolo Creminelli, Or Hershkovits, Leonardo Senatore, Andras Vasy
Summary: Using Mean Curvature Flow methods, this study examines 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying energy conditions, and spatial slices that can be foliated by 2-dimensional surfaces. It is found that if these surfaces have non-positive Euler characteristic and the initial spatial slice is expanding everywhere, the spacetime will asymptotically become physically indistinguishable from de Sitter space on arbitrarily large regions of spacetime. This conclusion holds true even in the presence of initial arbitrarily-large density fluctuations.
ADVANCES IN MATHEMATICS
(2023)
Article
Computer Science, Software Engineering
Kazuki Hayashi, Yoshiki Jikumaru, Makoto Ohsaki, Takashi Kagaya, Yohei Yokosuka
Summary: P-CMC surfaces are obtained through the mean curvature flow (MCF) and can be seen as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces. The MCF is formulated as the negative gradient flow of the energy functional for continuous surfaces, and then discretized to determine the change in vertex positions of triangular meshes on the surface and along the internal boundaries between patches. Numerical examples demonstrate that the proposed method enables multiple patch surfaces to approximately achieve the desired mean curvatures, which expands the options for shape design using CMC surfaces.
COMPUTER AIDED GEOMETRIC DESIGN
(2023)
Article
Mathematics, Applied
Sigurd Angenent, Simon Brendle, Panagiota Daskalopoulos, Natasa Sesum
Summary: This paper studies compact ancient solutions to the three-dimensional Ricci flow that are kappa-noncollapsed. It is proven that such solutions are either a family of shrinking round spheres or have a unique asymptotic behavior ast -> - infinity, which is described. This analysis is particularly applicable to the ancient solution constructed by Perelman.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Tim Binz, Balazs Kovacs
Summary: An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, with error estimates and convergence proof provided. The algorithm combines evolving surface finite elements and linearly implicit backward difference formulae for time integration, with stability analysis independent of geometric arguments.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Rafael Lopez, Alvaro Pampano
Summary: This study classifies and describes cylindrical surfaces in the Euclidean space with mean curvature as a nth-power of the distance to a reference plane, focusing on the generating curves known as n-elastic curves. It provides a variational characterization of these curves as critical points of a curvature energy, leading to a full description of the curves with closed curves observed in some specific cases.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)