Our work is developed within the field of the discovery of the influence that curvature (in its different extrinsic and intrinsic facets) has on the geometry, topology and analysis of a manifold and, reciprocally, in the study of the possible determination of curvature by these properties, and in the study of some of these properties motivated by physical problems. Specifically, we focus on: the study of Ricci-flat Kähler metrics on tangent bundles, twists by Higgs fields, some special curvatures of Wolf spaces, T-duality, the motion (and the consequent production of singularities) of a submanifold by different extrinsic geometric flows and approximations to the search for weak solutions of the Ricci flow and some of its variants to study peculiar geometric structures, the study of certain types of homogeneous spaces, the determination of certain geometric properties of a closed space by the spectrum of the Laplacian. It is worth highlighting the applicability of some of these problems to physics (unified theories), the formation of stable surfaces in some materials, and tomography. More specifically, we set the following specific objectives:
- Determine all Ricci-flat Kahler metrics on the tangent bundle of rank-2 symmetric spaces and Hermitian symmetric spaces.
- Find new families of G-invariant metrics on the spherical bundles indicated in the introduction, thus converting them into homogeneous Riemannian manifolds that are less rigid than the Sasaki metric.
- Study the general class of manifolds (possibly Kaehlerian) that we can relate by a class of generalized twists in which the compatibility conditions on the connection and curvature of the bundle are replaced by the canonical connection and the Higgs field equation.
- Study the extension of Chow & Yang's results to a hypothetical generalization of Gray's result and, in particular, study the properties of quaternionic bisectional curvature in Wolf spaces.
- Analyze this Strominger-Yau-Zaslow construction of mirror symmetry in a series of interesting cases: first, on semidirect products G X R^n, and, second, apply these techniques to the case of general nonvarieties.
- Advance the study of EBCV geometries and the search for a broader class of 7-dimensional manifolds in which this family can play a special role.
- Study of the rigidity of Angenent tori and calculation of the first eigenvalue of the Laplacian associated with the Gaussian density on these tori.
- Obtain Reilly-type bounds on the first eigenvalue of a Laplacian with density on submanifolds of hyperbolic space.
- Study the (in)stability of the Clifford torus under the VPMCF on the S^3 sphere first and, once understood, conduct similar studies in higher dimensions.
- Study of the evolution of non-embedded C^2 Lagrangian tori contained in a sphere, seeking to find type II singularities.
- Contribute to the study of higher codimension and/or boundary translation solitons.
- Study of mean curvature flow in hyperbolic space with Gaussian density.
- Use one of the transverse flows to a foliation, or some other of this type to study properties of structures in which the existence of a foliation is essential, among the spaces in which to consider these flows would be the EBCV.
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