Yamabe problem
Introduction
The Yamabe problem is a fundamental question in the field of differential geometry, specifically concerning the study of Riemannian manifolds. It involves finding a metric conformally equivalent to a given Riemannian metric on a compact manifold such that the resulting metric has constant scalar curvature. The problem is named after the Japanese mathematician Hidehiko Yamabe, who initially proposed the problem in 1960. The resolution of the Yamabe problem has profound implications in the study of geometric analysis and has connections to general relativity, conformal geometry, and partial differential equations.
Historical Background
The Yamabe problem was first formulated by Hidehiko Yamabe, who attempted to prove that on any compact Riemannian manifold, there exists a conformal metric with constant scalar curvature. However, Yamabe's original proof contained a critical error. The problem was subsequently addressed by several mathematicians, including Neil Trudinger, Thierry Aubin, and Richard Schoen, who collectively provided a complete solution over the course of several decades. Trudinger corrected Yamabe's error in 1968, Aubin made significant contributions in the 1970s, and Schoen completed the solution in 1984 by addressing the remaining cases.
Mathematical Formulation
The Yamabe problem can be formally stated as follows: Given a compact Riemannian manifold \((M, g)\), find a metric \(\tilde{g}\) conformally equivalent to \(g\) such that the scalar curvature \(R_{\tilde{g}}\) is constant. A metric \(\tilde{g}\) is said to be conformally equivalent to \(g\) if there exists a smooth positive function \(u\) on \(M\) such that \(\tilde{g} = u^{\frac{4}{n-2}}g\), where \(n\) is the dimension of the manifold.
The scalar curvature \(R_{\tilde{g}}\) of the conformally transformed metric \(\tilde{g}\) is given by the equation:
\[ R_{\tilde{g}} = u^{-\frac{n+2}{n-2}}(-c_n \Delta_g u + R_g u), \]
where \(c_n = \frac{4(n-1)}{n-2}\) and \(\Delta_g\) is the Laplace-Beltrami operator associated with the metric \(g\).
Solution and Techniques
The solution to the Yamabe problem involves several advanced techniques from geometric analysis and the theory of partial differential equations. The problem can be reduced to solving a nonlinear elliptic partial differential equation of the form:
\[ -c_n \Delta_g u + R_g u = \lambda u^{\frac{n+2}{n-2}}, \]
where \(\lambda\) is a constant that represents the constant scalar curvature of the new metric.
Conformal Invariance and Sobolev Inequalities
One of the key insights in solving the Yamabe problem is the use of conformal invariance properties and Sobolev inequalities. The Sobolev embedding theorem plays a crucial role in understanding the behavior of functions under conformal transformations. The Yamabe problem is closely related to the Sobolev inequality, which provides bounds on the integrability of functions and their derivatives.
The Role of Positive Mass Theorem
Richard Schoen's resolution of the Yamabe problem for all cases relied heavily on the positive mass theorem, a result in general relativity and differential geometry that asserts the positivity of the total mass of asymptotically flat manifolds. Schoen used this theorem to handle the case of manifolds with zero or negative scalar curvature, which was the last remaining obstacle in solving the Yamabe problem.
Applications and Implications
The resolution of the Yamabe problem has significant implications in several areas of mathematics and physics. In conformal geometry, it provides a method for understanding the structure of manifolds and their conformal classes. In general relativity, the Yamabe problem is related to the study of initial data sets for the Einstein equations, where the scalar curvature plays a role in the constraint equations.
Furthermore, the techniques developed to solve the Yamabe problem have been applied to other problems in geometric analysis, such as the study of Einstein metrics and the Ricci flow. The Yamabe problem also serves as a prototype for other geometric problems involving the modification of metrics to achieve desired curvature properties.
Generalizations and Related Problems
The Yamabe problem has inspired several generalizations and related problems in differential geometry. One such generalization is the prescribed scalar curvature problem, where the goal is to find a metric conformally equivalent to a given metric with a specified scalar curvature function. This problem is more challenging and remains an active area of research.
Another related problem is the study of Yamabe flow, a geometric flow that deforms the metric in the direction of its scalar curvature. The Yamabe flow is analogous to the Ricci flow and has been studied for its applications in understanding the geometry and topology of manifolds.