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Mathematics
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Advances in the Calculus of Variations and Geometry
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Advances in the Calculus of Variations and Geometry

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (15 September 2022) | Viewed by 3785

Special Issue Editor

Prof. Dr. Bennett Palmer

E-Mail Website
Guest Editor
Department of Mathematics, Idaho State University, Pocatello, ID 83209, USA
Interests: variational theory of surfaces in Euclidean space, especially in equilibrium surfaces for anisotropic surface energies

Special Issue Information

Dear Colleagues,

Th Special Issue “Advances in the Calculus of Variations and Geometry” will allow for publications of quality research involving geometric applications of variational theory.

From its inception, the subject of Differential Geometry has been inexorably linked with the Calculus of Variations through the study of geodesics. Interest in minimal surfaces, which began with the classical works of Riemann, Weierstrass, Schwarz, and others, has persisted into the the present time and has produced a wide range of exciting results.Generalizations of minimal surfaces such as higher-dimension minimal submanifolds, constant mean curvature submanifolds, and harmonic maps have developed into important research areas. Closely related to the variational theory of submanifolds are the curvature flows which produce equilibria through an evolutionary process. Widespread applications of geometric variational problems can be found not only in other areas of Mathematics such as Topology, but also in the physical sciences, particularly in General Relativity and Materials Science.

Colleagues are invited to submit papers concerning variational theory in Geometry and its applications in the physical sciences. These may include but are not limited to the subjects mentioned above.

Prof. Dr. Bennett Palmer
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Geodesics
  • Minimal submanifolds
  • Harmonic maps
  • Curvature flows
  • Geometric measure theory
  • Min-max constructions
  • Differential Geometry
  • Calculus of Variations
  • Topology

Published Papers (2 papers)

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Research

18 pages, 473 KiB  
Article
Existence and Uniqueness of a Curve with Both Minimal Length and Minimal Area
by Ariel Fuxman and Shai Gul
Mathematics 2022, 10(21), 4061; https://doi.org/10.3390/math10214061 - 1 Nov 2022
Viewed by 1305
Abstract
Consider the family of generalized parabolas {y=−axr+c|a,r,c,x>0,risafixedconstant} that pass through a given point in the first quadrant (and [...] Read more.
Consider the family of generalized parabolas {y=−axr+c|a,r,c,x>0,risafixedconstant} that pass through a given point in the first quadrant (and hence, depend on one parameter only). Find the parameter values for which the piece of the corresponding parabola in the first quadrant either encloses a minimum area, or has a minimum length. We find a sufficient condition under which given the fixed point, the area minimizing curve and the length minimizing curve coincide. The problem led us to a certain implicit function and we explored its asymptotic behavior and convexity. Full article
(This article belongs to the Special Issue Advances in the Calculus of Variations and Geometry)
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39 pages, 3197 KiB  
Article
The Existence of rG Family and tG Family, and Their Geometric Invariants
by Norio Ejiri and Toshihiro Shoda
Mathematics 2020, 8(10), 1693; https://doi.org/10.3390/math8101693 - 2 Oct 2020
Cited by 3 | Viewed by 1835
Abstract
In the 1990s, physicists constructed two one-parameter families of compact oriented embedded minimal surfaces in flat three-tori by using symmetries of space groups, called the rG family and tG family. The present work studies the existence of the two families via the period [...] Read more.
In the 1990s, physicists constructed two one-parameter families of compact oriented embedded minimal surfaces in flat three-tori by using symmetries of space groups, called the rG family and tG family. The present work studies the existence of the two families via the period lattices. Moreover, we will consider two kinds of geometric invariants for the two families, namely, the Morse index and the signature of a minimal surface. We show that Schwarz P surface, D surface, Schoen’s gyroid, and the Lidinoid belong to a family of minimal surfaces with Morse index 1. Full article
(This article belongs to the Special Issue Advances in the Calculus of Variations and Geometry)
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