Article
Multidisciplinary Sciences
Masaharu Nagayama, Harunori Monobe, Koya Sakakibara, Ken-Ichi Nakamura, Yasuaki Kobayashi, Hiroyuki Kitahata
Summary: In this study, a mathematical model of self-propelled objects based on the Allen-Cahn type phase-field equation is proposed. By combining it with the equation for the concentration of surfactant, the model is able to handle both shape change and motion of self-propelled objects. The model can represent both deformable and solid objects by controlling a single parameter. Moreover, it is demonstrated that the phase-field based model can be reduced to a free boundary model by taking the singular limit, which gives a physical interpretation to the model.
SCIENTIFIC REPORTS
(2023)
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
Kyeongsu Choi, Panagiota Daskalopoulos
Summary: This paper establishes the long time existence of complete non-compact weakly convex and smooth hypersurfaces evolving by the Q(k)-flow. The maximum existence time is shown to depend on the dimension of a vector space, and the paper discusses the conditions for the existence of solutions in different dimensions.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
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
Stephen Lynch
Summary: The study demonstrates that convex ancient solutions of mean curvature flow with Type I curvature growth are limited to being either spherical, cylindrical, or planar. It also proves a similar statement for flows described by a specific class of curvature functions that are convex or concave in the second fundamental form. These results do not require the assumption of non-collapsing interior.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2022)
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
Simon Brendle, Panagiota Daskalopoulos, Keaton Naff, Natasa Sesum
Summary: This paper studies the classification of ancient kappa-solutions to n-dimensional Ricci flow on Sn, extending the previous results in three dimensions. The study shows that such solutions are either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2023)
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
Mathematics
Theodora Bourni, Martin Reiris
Summary: The paragraph explains the construction of a slingshot, which is a compact and embedded solution for curve shortening flow. It originates from a non-compact curve and exists for a finite time.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Alexander Mramor, Alec Payne
Summary: The article discusses embedded ancient and eternal solutions related to mean curvature flow, providing examples in the study of unstable minimal hypersurfaces. These solutions are mean convex yet nonconvex, and are not affected by rigid motions or homotheties.
MATHEMATISCHE ANNALEN
(2021)
Article
Mathematics, Applied
Sebastian Hensel, Maximilian Moser
Summary: In this study, we extend the recent rigorous convergence result of the Allen-Cahn equation towards evolution by mean curvature flow with constant contact angle. We manage to remove the perturbative assumption on the contact angle being close to 90 degrees, and obtain more general convergence rate results. Our proof deviates from the popular strategies and instead relies on a relative entropy technique.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
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
Tobias Holck Colding, William P. I. I. I. I. Minicozzi
Summary: This article discusses the smoothing properties of parabolic geometric flows and the classification of singularities. By incorporating the dynamical properties of the flow, long-term smoothing for generic initial conditions can be achieved. In a specific case, it is shown that the dynamics of singularities can be categorized as the simplest possible.
JOURNAL OF DIFFERENTIAL GEOMETRY
(2021)
Article
Mathematics
Zhou Gang
Summary: In this study, we examine the formation of generic singularities under mean curvature flow by combining different approaches and results. We obtained detailed information when the initial hypersurfaces are close to cylinders and also included all the generic blowups from the results by Colding and Minicozzi. We focused on the cases where the rescaled flow converges to the cylinder S-1 x R-3 and extended the region controlled by Colding and Minicozzi to provide a finer description of the singularity's neighborhood.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics
Jake P. Solomon, Amitai M. Yuval
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2017)
Article
Mathematics
Yoel Groman, Jake P. Solomon
JOURNAL OF SYMPLECTIC GEOMETRY
(2016)
Article
Mathematics, Applied
Raz Kupferman, Michael Moshe, Jake P. Solomon
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2015)
Article
Mathematics
Peter Albers, Urs Frauenfelder, Jake P. Solomon
COMMENTARII MATHEMATICI HELVETICI
(2014)
Article
Mathematics
Paul Seidel, Jake P. Solomon
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2012)
Article
Mathematics
Jake P. Solomon
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2014)
Article
Mathematics
Yoel Groman, Jake P. Solomon
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2014)
Article
Mathematics
Raz Kupferman, Jake P. Solomon
JOURNAL OF FUNCTIONAL ANALYSIS
(2014)
Article
Mathematics
Jake P. Solomon
MATHEMATISCHE ANNALEN
(2013)
Article
Mathematics
Jake P. Solomon, Misha Verbitsky
COMPOSITIO MATHEMATICA
(2019)
Article
Mathematics
Jake P. Solomon
ADVANCES IN MATHEMATICS
(2020)
Correction
Mathematics
Jake P. Solomon, Sara B. Tukachinsky
Summary: The affiliations of authors were erroneously published due to a processing error.
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics
Jake P. Solomon, Sara B. Tukachinsky
Summary: This research introduces a new method for defining genus zero open Gromov-Witten invariants with boundary constraints, no longer restricted by the need for Lagrangian to be fixed by an anti-symplectic involution. By utilizing the technique of bounding chains and gauge equivalence classes of bounding chains, invariants can be defined and calculated effectively.
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics
Yanir A. Rubinstein, Jake P. Solomon
ADVANCES IN MATHEMATICS
(2017)