This chapter gives a brief survey of continuous time finance. We categorize diffusion models according to the nature of their volatility coefficient. Models whose volatility coefficient does not exhibit randomness are treated in Sect. 3.1. Models whose volatility coefficient follows a stochastic process are discussed in Sect. 3.2.

Continuous Time Finance

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.

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Manifolds with 1/4-pinched curvature are space forms

Topology and Geometry

/pdf/kahler-einstein-metrics-on-fano-manifolds-ii-limits-with-58i5d1zlwn.pdf

Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2

Nonlinear elliptic and parabolic equations of the second order

Manifolds with 1/4-pinched Curvature are Space Forms

It is well known that the PDE (1) has at least one positive solution for any choice of (M, g). If n > 6 and (M, g) is not locally conformally flat, this follows from results of T. Aubin [3]. The remaining cases were solved by R. Schoen using the positive mass theorem [16]. Solutions to (1) are not usually unique. As an example, consider the product metric on 51(L) x 5n_1(l). If L is sufficiently small, then the Yamabe PDE has a unique solution. On the other hand, there are many non-minimizing solutions if L is large. D. Pollack [14] has used gluing techniques to construct high energy solutions on more general background manifolds: given any conformai class with positive Yamabe constant and any positive integer iV, there exists a new conformai class which is close to the original one in the C?-norm and contains at least N metrics of constant scalar curvature (see [14], Theorem 0.1). It is an interesting question whether the set of all solutions to the Yamabe PDE is compact (in the C2-topology, say). A well-known conjecture states that this should be true unless (M, p) is conformally equivalent to the round sphere (see [17], [18], [19]). This conjecture has been verified in low dimensions and in the locally conformally flat case: if (M, g) is locally conformally flat, compactness follows from work of R. Schoen [17], [18]. Moreover, Schoen proposed a strategy for proving the conjecture in the non-locally conformally flat case based on the Pohozaev identity.

/pdf/blow-up-phenomena-for-the-yamabe-equation-woshawai66.pdf

Blow-up phenomena for the Yamabe equation

We consider the Yamabe flow $\frac{\partial g}{\partial t} = -(R_g - r_g) \, g$ where $g$is a Riemannian metric on a compact manifold $M, R_g$ denotes its scalar curvature, and $r_g$ denotes the mean value of the scalar curvature. We prove convergence of the Yamabe flow if the dimension $n$ satisfies $3 \leq n \leq 5$ or the initial metric is locally conformally flat.

/pdf/convergence-of-the-yamabe-flow-for-arbitrary-initial-energy-tlv7ppk4m7.pdf

Convergence of the Yamabe flow for arbitrary initial energy

We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.

Simon Brendle

Papers

Manifolds with 1/4-pinched curvature are space forms

Manifolds with 1/4-pinched Curvature are Space Forms

Blow-up phenomena for the Yamabe equation

Convergence of the Yamabe flow for arbitrary initial energy

Constant mean curvature surfaces in warped product manifolds