Santiago R. Simanca

Bio: Santiago R. Simanca is an academic researcher from Courant Institute of Mathematical Sciences. The author has contributed to research in topics: Scalar curvature & Ricci curvature. The author has an hindex of 8, co-authored 18 publications receiving 360 citations. Previous affiliations of Santiago R. Simanca include University of Miami & Stony Brook University.

Extremal Kähler metrics and complex deformation theory

Abstract: Let (M, J, g) be a compact Kahler manifold of constant scalar curvature. Then the Kahler class [ω] has an open neighborhood inH 1,1 (M, ℝ) consisting of classes which are represented by Kahler forms of extremal Kahler metrics; a class in this neighborhood is represented by the Kahler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [ω] is “nondegenerate,” every small deformation of the complex manifold (M, J) also carries Kahler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kahler metrics on certain compact complex surfaces.

A K-energy characterization of extremal Kahler metrics

Abstract: We show that for any polarized compact Kaihler manifold, the extremal Kaihler metrics that represent the given cohomology class can be characterized as critical points of a suitably defined K-energy functional. Let (M, J, Q) be a polarized Kaihler manifold of complex dimension n. Thus, (M, J) is a complex manifold of Kaihler type, and Q is a cohomology class in H1'l that can be represented by the Kahler form of a Kaihler metric. One may hope to find a canonical metric that represents Q by studying critical points of a suitable Riemannian functional. Since the set Q+ of Kahler metrics of fixed Kahler class Q is parametriced by (an open set of) functions, Calabi [2, 3] proposed the functional Q+ RIl

The Yamabe problem for almost Hermitian manifolds

Abstract: The conformal class of a Hermitian metric g on a compact almost complex manifold (M2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature sJ. We ask if the conformal class of g carries a metric with constant sJ, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)sJ−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere\(\mathbb{S}^6 \) equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.

Strongly Extremal Kähler Metrics

Abstract: On a compact complex manifold (M, J) of the Kahler type, we consider the functional defined by the L2-norm of the scalar curvature with its domain the space of Kahler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kahler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kahler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP2.

Some Open Problems in Elasticity

Abstract: Some outstanding open problems of nonlinear elasticity are described The problems range from questions of existence, uniqueness, regularity and stability of solutions in statics and dynamics to issues such as the modelling of fracture and self-contact, the status of elasticity with respect to atomistic models, the understanding of microstructure induced by phase transformations, and the passage from three-dimensional elasticity to models of rods and shells Refinements are presented of the author’s earlier work Ball [1984a] on showing that local minimizers of the elastic energy satisfy certain weak forms of the equilibrium equations

Kähler geometry of toric varieties and extremal metrics

Abstract: A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on X, using only data on Δ In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn A construction, due to Calabi, of a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on is recast very simply and explicitly using Guillemin's approach Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kahler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kahler class

Extremal Kähler metrics and complex deformation theory

Abstract: Let (M, J, g) be a compact Kahler manifold of constant scalar curvature. Then the Kahler class [ω] has an open neighborhood inH 1,1 (M, ℝ) consisting of classes which are represented by Kahler forms of extremal Kahler metrics; a class in this neighborhood is represented by the Kahler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [ω] is “nondegenerate,” every small deformation of the complex manifold (M, J) also carries Kahler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kahler metrics on certain compact complex surfaces.

Kahler geometry of toric varieties and extremal metrics

Abstract: Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is derived. A construction, due to Calabi, of a 1-parameter family of extremal metrics of non-constant scalar curvature is recast very simply and explicitly. Finally, a curious combinatorial formula for convex polytopes, that follows from the relation between the total integral of the scalar curvature and the wedge product of the first Chern class with a suitable power of the Kahler class, is presented.