Final Report Summary - GSCRM (Elliptic Submanifolds in Hyperbolic Geometry)
Project context
The project was subdivided into three interrelated subprojects which were then divided under two headings: Foliations of Hyperbolic Manifolds, and 3-dimensional hyperbolic space. In earlier work, the applicant has already shown how geometrically interesting familes of constant Gaussian curvature surfaces may be associated to families of (more or less) holomorphic functions on the complex plane. The area and volumes of these surfaces are finite, and thus yield functionals over the corresponding moduli space of holomorphic functions. The applicant believes that these functionals can provide tools for the study of Teichmüller theory. It is believed that these functionals are related to, or generate, geometric structures (such as symplectic structures) over these moduli spaces of holomorphic functions. Moreover, at the interdisciplinary level, these objects provide a toy model for understanding the physical theory of quantum strings, and knowledge of their properties would thus yield insight into potential applications of this model.
Project results
In work in preparation, the researcher has clarified the relationship between these constant Gaussian curvature surfaces and their corresponding families of holomorphic functions, showing that all such surfaces are complete, proper and of finite area. Moreover, all surfaces of constant Gaussian curvature in hyperbolic space which are complete, proper and of finite area arise from holomorphic functions in this manner. In addition, the area of these surfaces depends only their topology and thus do not provide interesting functionals over their Teichmüller space.
Project conclusions
The volume 'contained' by these surfaces may yet provide interesting functions over this Teichmüller space. In addition, concerning the application of these objects as a toy model for Quantum String Theory, in discussion with Edward Witten, he agreed that they did appear interesting even though it was not immediately clear precisely what that application may be.
Hyperbolic ends (3- or higher-dimensional): these arise as fundamental objects in the study of hyperbolic geometry, since they are the leftover parts of quasi-Fuchsian hyperbolic manifolds when the so-called Nielsen kernel is removed. They are also related to the study of flat conformal structures (FCSs). In earlier work, the applicant has shown how constant special Lagrangian (SL) curvature hypersurfaces may be constructed inside a large family of hyperbolic ends. There remain however many important cases where the existence of hypersurfaces of constant SL curvature is still not proven. Moreover, in the 3-dimensional case, these hypersurfaces yield continuous curves inside the moduli space of FCSs, the geometric properties of which constitutes an interesting study, closely related to existing work by Tanigawa.
The researcher has established stronger existence results and applications thereof to the moduli space. This work appears in Smith G., Moduli of Flat Conformal Structures of Hyperbolic Type, Geom. Dedicata, 154, no. 1, (2011), 4780. The geometric structure of the curves inside Teichmüller space generated by these hypersurfaces (relating to the work of Tanigawa) remains an interesting object of study.
Special Lagrangian curvature: Special Lagrangian (SL) curvature is a concept developed by the applicant which generalises the concept of Gaussian curvature of smooth, convex hypersurfaces. The researcher has established existence results and has fully understood the case of manifolds with boundary. This work appears in Smith G., The Non-Linear Dirichlet Problem in Hadamard Manifolds, arXiv:0908.3590; and Smith G., The non-linear Plateau problem in non-positively curved manifolds, to appear in Trans. Amer. Math. Soc. These results have opened up new research directions, leading to the realisation by the researcher of previously unexpected objectives, to appear in the following papers: Smith G., Compactness results for immersions of prescribed Gaussian curvature I - analytic aspects, to appear in Adv. Math.; Smith G., Compactness results for immersions of prescribed Gaussian curvature II - geometric aspects, arXiv:1002.2982; Clarke A., Smith G., The Perron Method and the Non-Linear Plateau Problem, to appear in Geom. Dedicata; Smith G., The Plateau problem for general curvature functions, arXiv:1008.3545. In addition, these results have opened the way to topological approaches to the study of these problems. Results to date appear in Smith G., Constant curvature hyperspheres and the Euler Characteristic, arXiv:1103.3235; and Rosenberg H., Smith G., Degree Theory of Immersed Hypersurfaces, arXiv:1010.1879.
The researcher has also studied the application of Morse theoretic techniques to the problem of existence of hypersurfaces of constant curvature, the first stages of which are outlined in work in progress as well as the following paper: Smith G., Eternal forced mean curvature flows I - A compactness result. Further work in progress treats the refinement of these degree theoretic techniques to the study of the specific case of real hypersurfaces in 2-dimensional complex projective space as well as the study of extremal domains in convex subsets of manifolds of non-negative Ricci curvature. Expertise gained in the field of Morse theory of Banach manifolds have also allowed the research to obtain elegant results concerning solutions of the Allen-Cahn equation. These results have led the researcher to apply the techniques of differential topology in infinite dimensions to the study of these problems. This has opened up a wide range of research possibilities which we expect to yield many more interesting results over the next few years.
The project was subdivided into three interrelated subprojects which were then divided under two headings: Foliations of Hyperbolic Manifolds, and 3-dimensional hyperbolic space. In earlier work, the applicant has already shown how geometrically interesting familes of constant Gaussian curvature surfaces may be associated to families of (more or less) holomorphic functions on the complex plane. The area and volumes of these surfaces are finite, and thus yield functionals over the corresponding moduli space of holomorphic functions. The applicant believes that these functionals can provide tools for the study of Teichmüller theory. It is believed that these functionals are related to, or generate, geometric structures (such as symplectic structures) over these moduli spaces of holomorphic functions. Moreover, at the interdisciplinary level, these objects provide a toy model for understanding the physical theory of quantum strings, and knowledge of their properties would thus yield insight into potential applications of this model.
Project results
In work in preparation, the researcher has clarified the relationship between these constant Gaussian curvature surfaces and their corresponding families of holomorphic functions, showing that all such surfaces are complete, proper and of finite area. Moreover, all surfaces of constant Gaussian curvature in hyperbolic space which are complete, proper and of finite area arise from holomorphic functions in this manner. In addition, the area of these surfaces depends only their topology and thus do not provide interesting functionals over their Teichmüller space.
Project conclusions
The volume 'contained' by these surfaces may yet provide interesting functions over this Teichmüller space. In addition, concerning the application of these objects as a toy model for Quantum String Theory, in discussion with Edward Witten, he agreed that they did appear interesting even though it was not immediately clear precisely what that application may be.
Hyperbolic ends (3- or higher-dimensional): these arise as fundamental objects in the study of hyperbolic geometry, since they are the leftover parts of quasi-Fuchsian hyperbolic manifolds when the so-called Nielsen kernel is removed. They are also related to the study of flat conformal structures (FCSs). In earlier work, the applicant has shown how constant special Lagrangian (SL) curvature hypersurfaces may be constructed inside a large family of hyperbolic ends. There remain however many important cases where the existence of hypersurfaces of constant SL curvature is still not proven. Moreover, in the 3-dimensional case, these hypersurfaces yield continuous curves inside the moduli space of FCSs, the geometric properties of which constitutes an interesting study, closely related to existing work by Tanigawa.
The researcher has established stronger existence results and applications thereof to the moduli space. This work appears in Smith G., Moduli of Flat Conformal Structures of Hyperbolic Type, Geom. Dedicata, 154, no. 1, (2011), 4780. The geometric structure of the curves inside Teichmüller space generated by these hypersurfaces (relating to the work of Tanigawa) remains an interesting object of study.
Special Lagrangian curvature: Special Lagrangian (SL) curvature is a concept developed by the applicant which generalises the concept of Gaussian curvature of smooth, convex hypersurfaces. The researcher has established existence results and has fully understood the case of manifolds with boundary. This work appears in Smith G., The Non-Linear Dirichlet Problem in Hadamard Manifolds, arXiv:0908.3590; and Smith G., The non-linear Plateau problem in non-positively curved manifolds, to appear in Trans. Amer. Math. Soc. These results have opened up new research directions, leading to the realisation by the researcher of previously unexpected objectives, to appear in the following papers: Smith G., Compactness results for immersions of prescribed Gaussian curvature I - analytic aspects, to appear in Adv. Math.; Smith G., Compactness results for immersions of prescribed Gaussian curvature II - geometric aspects, arXiv:1002.2982; Clarke A., Smith G., The Perron Method and the Non-Linear Plateau Problem, to appear in Geom. Dedicata; Smith G., The Plateau problem for general curvature functions, arXiv:1008.3545. In addition, these results have opened the way to topological approaches to the study of these problems. Results to date appear in Smith G., Constant curvature hyperspheres and the Euler Characteristic, arXiv:1103.3235; and Rosenberg H., Smith G., Degree Theory of Immersed Hypersurfaces, arXiv:1010.1879.
The researcher has also studied the application of Morse theoretic techniques to the problem of existence of hypersurfaces of constant curvature, the first stages of which are outlined in work in progress as well as the following paper: Smith G., Eternal forced mean curvature flows I - A compactness result. Further work in progress treats the refinement of these degree theoretic techniques to the study of the specific case of real hypersurfaces in 2-dimensional complex projective space as well as the study of extremal domains in convex subsets of manifolds of non-negative Ricci curvature. Expertise gained in the field of Morse theory of Banach manifolds have also allowed the research to obtain elegant results concerning solutions of the Allen-Cahn equation. These results have led the researcher to apply the techniques of differential topology in infinite dimensions to the study of these problems. This has opened up a wide range of research possibilities which we expect to yield many more interesting results over the next few years.