The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the C…
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the s…
On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously pro…
We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-Émery curvature. The model…
In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of…
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establis…
The paper derives subgradient estimates for a specific nonlinear subparabolic equation on pseudo-Hermitian manifolds.
problem Deriving subgradient estimates for positive solutions to a nonlinear subparabolic equation on pseudo-Hermitian manifolds.
method Using the CR sub-Laplacian comparison property, the paper derives local subgradient estimates for positive solutions to the given equation.
result The paper establishes subgradient estimates for positive solutions to the nonlinear subparabolic equation.
For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider …
This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. A…
The main technical result of the paper is a Bochner type formula for the sub-laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(n)Sp(1) components of the …
We prove a CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Panietz operator which satisfies a Lichnerowicz type condition. We show that if the first positive eigenvalue of the…
Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. We prove the contact Riemannian version of…
Study solves sub-Laplacian equivalence on a specific Heisenberg group.
problem Contact equivalence problem for sub-Laplacians on the second Heisenberg group.
method Solves the contact equivalence problem for generalised sub-Laplacians on $\He^2$.
result Parameterises sub-Laplacians on $\He^2$ by R+. A version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(1)Sp(1) component of the qc-Ricci curvature on a compact seven dimensional quaternionic contact manifold is established. It is shown that in the case of a seven dimensional compact 3-Sasakian ma…
We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart S…
Maps commuting with sub-Laplacians on Carnot groups are conformal.
problem Characterizing maps preserving sub-Laplacians on sub-Riemannian Lie groups.
method Analyzing smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians.
result Sub-Laplacian determines the sub-Riemannian structure in Carnot groups.
New model explains visual illusions using sub-Riemannian geometry.
problem Mathematical modeling of Poggendorff-type visual illusions.
method Cortical-inspired sub-Riemannian model with sub-Laplacian evolution.
result Sub-Riemannian kernel enhances visual misperceptions and biases.
We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher…
Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
problem Characterizing the intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
method Construction and analysis of the intrinsic sub-Laplacian using Riemannian approximations and stochastic processes.
result The intrinsic sub-Laplacian is stochastically complete, ensuring the process does not hit characteristic points.
In this paper, we generalize the CR Obata theorem to a compact strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the Witten sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its applications, we obt…
Generalizes Cheeger inequality to Carnot-Carathéodory spaces.
problem Lower bounds on eigenvalues of Laplacians in complex spaces.
method Geometric approach, including Neumann and mixed boundary conditions.
result Concrete method to lower bound Cheeger constant.
We define a CR structure on a distinguished hyperplane in Cn+1 and the CR sub-Laplacian on this CR manifold. We also define symmetries of the CR sub-Laplacian in general and for this special case construct all of them using the ambient construction. Then we investigate the algebra structure of the symmetr…
The paper explores the index theory of sub-Laplacians on higher nilpotent Carnot manifolds.
problem Characterizing the index theory of sub-Laplacians on higher nilpotent Carnot manifolds.
method Analyzes the structure of hypoelliptic sub-Laplacian type operators and provides examples where the index theory is trivial.
result Provides examples where the index theory of sub-Laplacians is trivial in higher degrees of nilpotency.
Study spectral properties of sub-Laplacians in Carnot groups.
problem Spectral properties of sub-Laplacians in Carnot groups.
method Proved pure point spectrum and spectral gap; applied to small ball problem and heat content.
result Proved existence of spectral gap and pure point spectrum.
Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.
problem Analyzing sub-Riemannian Brownian motions and their radial processes.
method Application of Itô's formula and sub-Laplacian comparison theorems to prove stochastic completeness and eigenvalue estimates.
result Proved Cheng's type estimates for Dirichlet eigenvalues of sub-Riemannian metric balls.
Study on nonholonomic mechanics and sub-Finsler geometry.
problem Understanding nonholonomic mechanical systems and their geometric properties.
method Variational approach, sub-Finsler manifolds, nonholonomic sub-Finslerian structure.
result Existence and properties of extremals in nonholonomic mechanics.
In this paper we establish an analogue of the classical Lichnerowicz' theorem giving a sharp lower bound of the first non-zero eigenvalue of the sub-Laplacian on a compact seven-dimensional quaternionic contact manifold, assuming a lower bound of the qc-Ricci tensor, torsion tensor and its distinguished covariant deriv…
We consider different sub-Laplacians on a sub-Riemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in \cite{GordinaLaetsch2014a}. This operator is canonical with respec…
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components …
We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for…
We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian s…
We study the sub-Laplacian of the 15-dimensional unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the octonionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of a related sub-Laplacian. As a …
The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal s…
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjoestrand asymptotic law, and can be viewed as a CR version of Korevaar's bounds for Laplace eigenvalues of conformal metrics.
The study develops inequalities for Riemannian foliations without bundle-like assumptions.
problem Developing inequalities for Riemannian foliations without restrictive conditions.
method Bochner theory and Bakry-Emery calculus for horizontal Laplacians, derived explicit Bochner formulas, generalized curvature dimension inequalities.
result Established generalized curvature dimension inequalities for Riemannian foliations.
This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold M. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators natur…
CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ``conformally invariant powers of the Laplacian'' via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. …
We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure $\θ$ on the CR sphere S 2n+1 ⊂ C n+1 , achieves its maximum when $\θ$ is the standard contact form.
We prove a quaternionic contact versions of the Obata's sphere theorems. We show that if the first positive eigenvalue of the sub-Laplacian on a compact qc manifold of dimension bigger than seven takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard…
New stability theorems for H-type Carnot groups established.
problem Characterize H-type Carnot groups using stability theorems.
method Introduced H-type deviation, computed for families, established new characterizations.
result H-type Carnot groups are uniquely characterized by specific conditions.
Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian
problem Sub-Riemannian geometry and eigenvalues of sub-Laplacian
method Embedding manifold into Hilbert space using eigenfunctions
result Defined spectral distance between sub-Riemannian manifolds
Formula calculates manifold Euler characteristic from curvature.
problem Computing Euler characteristic from curvature invariants.
method Used hypoelliptic sub-Laplacian on forms.
result Euler characteristic can be computed from curvature invariants.
We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…
In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q0∈M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0∞(M∖{q0}) is essentially self-adjoint. A particular …
Study examines harmonic functions in sub-Riemannian and RCD settings.
problem Characterizing harmonic functions in sub-Riemannian and RCD settings.
method Analyzes weak and strong asymptotically mean value harmonic functions.
result Weakly amv-harmonic functions are equivalent to harmonicity in Carnot groups.