By n/a, Lizhen Ji (University of Michigan), Peter Li (University of California, Irvine), Richard Schoen (Stanford University), Leon Simon (Stanford University)

Geometric research combines differential equations with differential geometry. a tremendous point of geometric research is to strategy geometric difficulties via learning differential equations. along with a few identified linear differential operators equivalent to the Laplace operator, many differential equations bobbing up from differential geometry are nonlinear. a very vital instance is the Monge-Amper? equation. functions to geometric difficulties have additionally encouraged new equipment and strategies in differential equations. the sector of geometric research is vast and has had many notable functions.

This instruction manual of geometric research the 1st of the 2 to be released within the ALM sequence offers introductions and survey papers treating very important issues in geometric research, with their functions to similar fields. it may be used as a reference by means of graduate scholars and through researchers in comparable components. desk of contents Numerical Approximations to Extremal Metrics on Toric Surfaces (R. S. Bunch, Simon okay. Donaldson) K?hler Geometry on Toric Manifolds, and a few different Manifolds with huge Symmetry (Simon okay. Donaldson) Gluing structures of particular Lagrangian Cones (Mark Haskins, Nikolaos Kapouleas) Harmonic Mappings (J?rgen Jost) Harmonic capabilities on entire Riemannian Manifolds (Peter Li) Complexity of recommendations of Partial Differential Equations (Fang Hua Lin) Variational ideas on Triangulated Surfaces (Feng Luo) Asymptotic constructions within the Geometry of balance and Extremal Metrics (Toshiki Mabuchi) reliable consistent suggest Curvature Surfaces (William H. Meeks III, Joaqu?n P?rez, Antonio Ros) A basic Asymptotic Decay Lemma for Elliptic difficulties (Leon Simon) Uniformization of Open Nonnegatively Curved K?hler Manifolds in larger Dimensions (Luen-Fai Tam) Geometry of Measures: Harmonic research Meets Geometric degree conception (Tatiana Toro) Lectures on suggest Curvature Flows in better Codimensions (Mu-Tao Wang) neighborhood and worldwide research of Eigenfunctions on Riemannian Manifolds (Steve Zelditch) Yau's type of Schwarz Lemma and Arakelov Inequality On Moduli areas of Projective Manifolds (Kang Zuo)

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7 if we can show that (24) sup [(1 tE[O,T) - t) supscaly(0, < oo. SZ To prove (24), we argue by contradiction. Suppose that (24) is false. We define a sequence of times tk E [0, T) by tk = inf { t E [O, T) : (1 - t) sup scal g(t) > 2k } . S2 In the following, we will choose k sufficiently large so that tk > 0. For each k, we choose a point Pk E S2 where the scalar curvature of g(tk) attains its maximum. This implies scaly(tk) (Pk) = sup scalg(tk) S2 2k 1 - tk 4. 3, there exists a uniform constant N > 1 such that sup Idscals(tk) 12

12. Let X, Y, Z, W be fixed vector fields on M. Then a R(X, Y, Z, W) at = (zR)(X, Y, Z, W) + Q(R)(X, Y, Z, W) n n Ric(X, CO R(ek,Y, Z, W) - L Ric(Y, CO R(X, ek, Z) W) k= n n Ric(Z, CO R(X, Y, ek, W) - L Ric(W, CO R(X, Y, Z) ek). k=1 Let E be the pull-back of the tangent bundle TM under the projection M x (0, T) -+ M, (p, t) E p. In other words, the fiber of E over a point (p, t) E M x (0, T) is given by E(p,t) = TpM. There is a natural connection D on E, which extends the Levi-Civita connection on TM.

For each point (p, t) E M x (0, T) , we can find a linear isometry from Rn to E(p,t). This induces a linear isometry from 6B (IRn) to `6B (E(p,t)) . Let F(p,t) C %B (E(p,t)) be the image of the set F C WB (Rn) under this linear isometry. Since F is 0(n)-invariant, the set F(p,t) is well defined; that is, F(p,t) is independent of the choice of the linear isometry from IRn to E(p,t). 9. Assume that F C WB (Rn) is closed, convex, and O(n)invariant. Moreover, let M be a compact manifold of dimension n, and let g (t), t c [0, T), be a solution to the Ricci low on M.