|
Willmore Conjecture
In differential geometry, the Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English mathematician Tom Willmore, who conjectured it in 1965. A proof by Fernando Codá Marques and André Neves was announced in 2012 and published in 2014. Willmore energy Let ''v'' : ''M'' → R3 be a smooth immersion of a compact, orientable surface. Giving ''M'' the Riemannian metric induced by ''v'', let ''H'' : ''M'' → R be the mean curvature (the arithmetic mean of the principal curvatures ''κ''1 and ''κ''2 at each point). In this notation, the ''Willmore energy'' ''W''(''M'') of ''M'' is given by : W(M) = \int_M H^2 \, dA. It is not hard to prove that the Willmore energy satisfies ''W''(''M'') ≥ 4''π'', with equality if and only if ''M'' is an embedded round sphere. Statement Calculation of ''W''(''M'') for a few examples suggests that there should be a better bound than ''W' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willmore Conjecture - Perfect Doughnut
Willmore may refer to: People * Ben Willmore, American photographer and technology writer * Henrietta Willmore (1842–1938), Australian musician and suffragette * Ian Willmore (1958–2020), British activist * James Tibbits Willmore (1800–1863), British engraver * Jeff Willmore (born 1954), Canadian artist * Micheál Mac Liammóir (1899–1978), British–Irish actor, writer, painter etc. * Norman Willmore (1909–1965), politician in Alberta, Canada * Patrick Willmore (1921–1994), British seismologist * Thomas Willmore (1919–2005), English geometer * William Erwin Willmore (either 1844 or 1845–1901), English-born American teacher and the founder of a colony Other * The Willmore, an apartment building in California * Willmore Wilderness Park, Alberta, Canada See also * Willmore energy in differential geometry ** Willmore conjecture In differential geometry, the Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point. Discussion At each point ''p'' of a differentiable surface in 3-dimensional Euclidean space one may choose a unit '' normal vector''. A '' normal plane'' at ''p'' is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures for different normal planes at ''p''. The principal curvatures at ''p'', denoted ''k''1 and ''k''2, are the maximum and minimum values of this curvature. Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle. The curvature is taken to be positive i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus). Properties The sections of the surface of revolution made by planes through the axis are called ''meridional sections''. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles. Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tube Torus
Tube or tubes may refer to: * ''Tube'' (2003 film), a 2003 Korean film * ''The Tube'' (TV series), a music related TV series by Channel 4 in the United Kingdom * "Tubes" (Peter Dale), performer on the Soccer AM television show * Tube (band), a Japanese rock band * Tube & Berger, the alias of dance/electronica producers Arndt Rörig and Marco Vidovic from Germany * The Tube Music Network, a music video network that operated between 2006 and 2007 * The Tubes, a San Francisco-based band, popular in the 1970s and 1980s Other media * Tube, a freeware game for MS-DOS computers from Bullfrog Productions * ''TUBE.'', an online magazine about visual and performing arts, founded in 2012 in Sacramento, California * Series of tubes, an analogy for the Internet used by United States Senator Ted Stevens * Picture tube, term in Paint Shop Pro software for a small digital image with no background * YouTube, a video sharing website Science, technology, and mathematics Construction and mechanic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Huffington Post
''HuffPost'' (formerly ''The Huffington Post'' until 2017 and sometimes abbreviated ''HuffPo'') is an American progressive news website, with localized and international editions. The site offers news, satire, blogs, and original content, and covers politics, business, entertainment, environment, technology, popular media, lifestyle, culture, comedy, healthy living, women's interests, and local news featuring columnists. It was created to provide a progressive alternative to the conservative news websites such as the Drudge Report. The site offers content posted directly on the site as well as user-generated content via video blogging, audio, and photo. In 2012, the website became the first commercially run United States digital media enterprise to win a Pulitzer Prize. Founded by Andrew Breitbart, Arianna Huffington, Kenneth Lerer, and Jonah Peretti, the site was launched on May 9, 2005 as a counterpart to the Drudge Report. In March 2011, it was acquired by AOL for US$315& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Morgan (mathematician)
Frank Morgan is an American mathematician and the Webster Atwell '21 Professor of Mathematics, Emeritus, at Williams College. He is known for contributions to geometric measure theory, minimal surfaces, and differential geometry, including the resolution of the double bubble conjecture. He was vice-president of the American Mathematical Society and the Mathematical Association of America. Morgan studied at the Massachusetts Institute of Technology and Princeton University, and received his Ph.D. from Princeton in 1977, under the supervision of Frederick J. Almgren Jr. He taught at MIT for ten years before joining the Williams faculty. Morgan is the founder of SMALL, one of the largest and best known summer undergraduate Mathematics research programs. In 2012 he became a fellow of the American Mathematical Society. Frank Morgan is also an avid dancer. He gained temporary fame for his work "Dancing the Parkway". Mathematical work He is known for proving, in collaboration wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almgren–Pitts Min-max Theory
In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's method for the construction of simple closed geodesics on the sphere, to allow the construction of embedded minimal surfaces in arbitrary 3-manifolds. It has played roles in the solutions to a number of conjectures in geometry and topology found by Almgren and Pitts themselves and also by other mathematicians, such as Mikhail Gromov, Richard Schoen, Shing-Tung Yau, Fernando Codá Marques, André Neves, Ian Agol, among others. Description and basic concepts The theory allows the construction of embedded minimal hypersurfaces though variational methods. In his PhD thesis Almgren proved that the m-th homotopy group of the space of flat k-dimensional cycles on a closed Riemannian manifold is isomorphic to the (m+k)-th dimensional homolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Wai-Kwong Li
Peter Wai-Kwong Li (born 18 April 1952) is a mathematician whose research interests include differential geometry and partial differential equations, in particular geometric analysis. After undergraduate work at California State University, Fresno, he received his Ph.D. at University of California, Berkeley under Shiing-Shen Chern in 1979. Presently he is Professor Emeritus at University of California, Irvine, where he has been located since 1991. His most notable work includes the discovery of the Li–Yau differential Harnack inequalities, and the proof of the Willmore conjecture in the case of non-embedded surfaces, both done in collaboration with Shing-Tung Yau. He is an expert on the subject of function theory on complete Riemannian manifolds. He has been the recipient of a Guggenheim Fellowship in 1989 and a Sloan Research Fellowship. In 2002, he was an invited speaker in the Differential Geometry section of the International Congress of Mathematicians in Beijing, where he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
