Riemannian geometry | Mathematical problems
The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metric fg has constant scalar curvature. By computing a formula for how the scalar curvature of fg relates to that of g, this statement can be rephrased in the following form: Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function φ on M, and a number c, such that Here n denotes the dimension of M, Rg denotes the scalar curvature of g, and ∆g denotes the Laplace-Beltrami operator of g. The mathematician Hidehiko Yamabe, in the paper , gave the above statements as theorems and provided a proof; however, discovered an error in his proof. The problem of understanding whether the above statements are true or false became known as the Yamabe problem. The combined work of Yamabe, Trudinger, Thierry Aubin, and Richard Schoen provided an affirmative resolution to the problem in 1984. It is now regarded as a classic problem in geometric analysis, with the proof requiring new methods in the fields of differential geometry and partial differential equations. A decisive point in Schoen's ultimate resolution of the problem was an application of the positive energy theorem of general relativity, which is a purely differential-geometric mathematical theorem first proved (in a provisional setting) in 1979 by Schoen and Shing-Tung Yau. There has been more recent work due to Simon Brendle, Marcus Khuri, Fernando Codá Marques, and Schoen, dealing with the collection of all positive and smooth functions f such that, for a given Riemannian manifold (M,g), the metric fg has constant scalar curvature. Additionally, the Yamabe problem as posed in similar settings, such as for complete noncompact Riemannian manifolds, is not yet fully understood. (Wikipedia).
Yamabe flow of asymptotically flat metrics - Yi Wang
Members' Colloquium Topic: Yamabe flow of asymptotically flat metrics Speaker: Yi Wang Affiliation: Johns Hopkins University Date: October 10, 2022 In this talk, we will discuss the behavior of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time exis
From playlist Mathematics
Bernd Ammann - Yamabe constants, Yamabe invariants, and Gromov-Lawson surgeries
In this talk I want to study the (conformal) Yamabe constant of a closed Riemannian (resp. conformal) manifold and how it is affected by Gromov-Lawson type surgeries. This yields information about Yamabe invariants and their bordism invariance. So far the talk gives an overview over older
From playlist Not Only Scalar Curvature Seminar
Problem #21 - Physics of Yo-Yo's
Problem #21 - Physics of Yo-Yo's
From playlist Bi-weekly Physics Problems
Some problems using Lagrange Multipliers for optimization. In this video there are some technical problems beginning at about 9:10. The first problem is worked entirely, but the 2nd problem is interrupted.
From playlist Calc3Exam3Fall2013
GeoGebra Link: https://www.geogebra.org/m/yvqwqk6h
From playlist Geometry: Challenge Problems
ESS1C - The History of the Earth
In this video Paul Andersen explains in more detail the history of the Earth. He shows how the history of the Earth is written in the rocks that are built up over time. Fossils allow us to compare different rock layers relative to one another and the absolute radiometric dating allows sc
From playlist Next Generation Science Standards
Infinite solutions of the singular Yamabe problem in spheres via Teichmüller theory - Paolo Piccione
Variational Methods in Geometry Seminar Topic: Infinite solutions of the singular Yamabe problem in spheres via Teichmüller theory Speaker: Paolo Piccione Affiliation: University of Sao Paulo Date: April 25, 2019 For more video please visit http://video.ias.edu
From playlist Variational Methods in Geometry
Existence and uniqueness of Green's function to a nonlinear Yamabe problem - Yanyan Li
Workshop on Geometric Functionals: Analysis and Applications Topic: Existence and uniqueness of Green's function to a nonlinear Yamabe problem Speaker: Yanyan Li Affiliation: Rutgers University Date: March 6, 2019 For more video please visit http://video.ias.edu
From playlist Mathematics
Claude LeBrun - Yamabe invariants, Weyl curvature, and the differential topology of 4-manifolds
The behavior of the Yamabe invariant, as defined in Bernd Ammann’s previous lecture, differs strangely in dimension 4 from what is seen in any other dimension. These peculiarities not only manifest themselves in the context of the usual scalar curvature, but also occur in connection with
From playlist Not Only Scalar Curvature Seminar
Conformal Invariants from Nodal Sets - Dmitry Jakobson
Dmitry Jakobson McGill University April 23, 2013 We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establi
From playlist Mathematics
Problem #3 - Swinging Pendulum
Problem #3 - Swinging Pendulum
From playlist Bi-weekly Physics Problems
Solution to Problem 59
From playlist Solutions to Bi-weekly Physics Problems
GeoGebra Resource: https://www.geogebra.org/m/vwy7mvgs
From playlist Geometry: Challenge Problems
GeoGebra Link: https://www.geogebra.org/m/ketkkfuj
From playlist Geometry: Challenge Problems
Variational Surface Cutting - SIGGRAPH 2018
Variational Surface Cutting. Nicholas Sharp and Keenan Crane. ACM Trans. on Graph. (2018) http://www.cs.cmu.edu/~kmcrane/Projects/VariationalCuts/paper.pdf This paper develops a global variational approach to cutting curved surfaces so that they can be flattened into the plane with low m
From playlist Research
problem #92 - Polarized Sunglasses
From playlist Bi-weekly Physics Problems
In this video Paul Andersen describes some of the properties of waves. He starts be identifying particles and waves as the only phenomenon that can transfer energy from place to place. He identifies the defining characteristics of waves; wavelength, frequency and amplitude. He defines s
From playlist Next Generation Science Standards
Problem #24 Circuit with Five Resistors
Problem #24 Circuit with Five Resistors
From playlist Bi-weekly Physics Problems
GPDE Workshop - Conformal invariants and nonlinear Yamabe problem on manifolds... - Sophie
Sophie Chen IAS/University of California, Berkeley February 24, 2009 For more videos, visit http://video.ias.edu
From playlist Mathematics