Analysis of sub-Riemannian structures and related operators

• Applicants: Professor Dr. Wolfram Bauer; Professorin Dr. Ines Kath
• Subject Area: Mathematics
• Term: from 2010 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 189396777

• Final Report Year: 2014

Final Report Abstract

Among the various geometries on manifolds sub-Riemannian structures have attracted a strong interest during the last years. Various problems in the area of mathematics, physics or even applied sciences may be reformulated as a connectivity problem under non-holonomic constraints and can be treated in the framework of sub-Riemannian geometry. The notion of a Riemannian manifold forms the basic concept of elliptic operator theory. As is well known there are close links between the spectral theory of the Laplacian and the underlying geometry. In analogy the sub-Riemannian structures considered here induce an intrinsic sub-Laplacian ∆sub which, in contrast to the Laplacian, is just a sub-elliptic operator but still has similar analytic properties. Moreover, in this project we consider sub-Riemannian analogues of the classical Dirac operator. Similar to the elliptic case one may ask to what extend the spectral theory of ∆sub provides some geometric information and, conversely, whether a study of the sub-Riemannian geometry (e.g. admissible curves, curvature, geodesics · · · ) helps to analyze analytical objects (e.g. the heat kernel, spectral zeta function · · · ). Not all manifolds are carrying a sub-Riemannian structure and the first aim of this project is to determine interesting example classes of what we call a trivializable sub-Riemannian structure. In all these cases the intrinsic sub-Laplacian can be expressed as a sum of squares of global vector fields (”sum of square operator”). More precisely, we classify trivializable sub-Riemannian structures on Euclidean spheres and compact nilmanifold. The induced sub-Laplacian has discrete spectrum and we provide its spectral analysis. In the case of the sub-Laplacian on an arbitrary step 2 compact nilmanifold it came as a surprise to us that its spectral zeta function has a rather simple distribution of singularities. Only one simple pole exists with residue having a certain invariance property. This is in contrast to the spectral zeta function of the Laplacian on the same type of manifold when extending the sub-Riemannian structure to a Riemannian structure. Our analysis is based on an explicit form of the heat kernel for the (sub)-Laplacian on step two nilpotent Lie groups. Such formulas are not known in the case of higher step groups. An approach to derive heat kernels of higher step operators only leads to an ”approximate solution”. However, we successfully could calculate the fundamental solutions of such operators and apply the result to the analysis of the so-called Kohn-Laplacian. Let (M, H, g) be a sub-Riemannian manifold and assume that we have a metric connection on H. We define and analyze a sub-Riemannian Dirac operator D on M and in particular we treat the case of nilmanifolds M = Γ\G. In the two step case we can calculate the spectrum of D for several examples. As it turns out D has pure point spectrum but different from the spectrum of the sub-Laplacian eigenvalues of infinite multiplicity appear. In a second part of the project we study Toeplitz operators acting on the standard weighted Bergman spaces over bounded symmetric domains (bsd) Ω ⊂ Cn . We aim to characterize compactness and boundedness of these operators that are uniform in the weight parameter. As was observed in earlier works the corresponding problem in the ”flat case” Ω = Cn involves properties of the heat transform of the operator symbol. In our setting of a bsd we use the so-called λ-Berezin transform (λ=weight parameter) of the symbol instead and prove that its supremum norm is dominated by the norm of the corresponding Toeplitz operator for a specific choice of weights. As was observed in the sequel, such estimates are of great use in the study of Toeplitz operators with uniformly continuous symbols with respect to the Bergman (hyperbolic) distance function.

Publications

• Compact Toeplitz operators for weighted Bergman spaces on bounded symmetric domains, Integr. Equ. Oper. Theory 70 (2011) no. 4, 569-582
  H. Issa
  
• An action function for a higher step Grushin operator, J. Geom. Phys. 62 (2012), no. 9, 1949-1976
  K. Furutani , C. Iwasaki , T. Kagawa
  
• Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal-Bargmann space, J. Math. Anal. Appl. 386 (2012), no. 1, 213-235
  W. Bauer , H. Issa
  
• Spectral zeta function of the sub-Laplacian on two step nilmanifolds, J. Math. Pures Appl. 97, (2012) 242-261
  W. Bauer , K. Furutani , C. Iwasaki
  
• pectra of sub-Dirac operators on certain nilmanifolds
  I. Kath , O. Ungermann
  
• Trivializable sub-Riemannian structures on spheres, Bull. Sci. Math. 137, (2013) 361-385
  W. Bauer , K. Furutani , and C. Iwasaki
  
• Fundamental solution of a higher step Grushin type operator. Advances in Mathematics Volume 271, 5 February 2015, Pages 188-234
  W. Bauer , K. Furutani , C. Iwasaki
  
• The inverse of a parameter family of degenerate operators and applications to the Kohn-Laplacian. Advances in Mathematics Volume 277, 4 June 2015, Pages 283-337
  W. Bauer , K. Furutani , C. Iwasaki