arXiv research

Rigidity of critical eigensections on spheres proven.

We derive a bound on the $L^{\infty}$-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with sma…

Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.

Exponential localization of eigensections for Bochner-Schrödinger operator.

The paper studies spectral analysis on complex spaces and finds explicit formulas for eigensections.

Let $Y$ be a compact, oriented 3-manifold with a contact form $a$ and a metric $ds^2$. Suppose that $F\to Y$ is a principal bundle with structure group $U(2) = SU(2)\times_{\pm1}S^1$ such that $F/S^1$ is the principal SO(3) bundle of orthonormal frames for $TY$. A unitary connection $A_0$ on the Hermitian line bundle $…

We introduce and study new spectral invariant of two elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depends on both the eigenvalues and the eigensections of the operators, which is a equal to the regularized number of created particles from the vacuum…

We introduce and study {\it new} relative spectral invariants of {\it two} elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the eigensections of these operators and contain much more information about geometry. We prov…

It is known that a compact symplectic manifold endowed with a prequantum line bundle can be embedded in the projective space generated by the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We show that the Fubini-Study forms induced by these embeddin…

Bounds on spectral gaps of hyperbolic 3-manifolds and orbifolds.

Study on $G_{2}$-instantons and Hermitian Yang-Mills connections, focusing on spectrum analysis.

Study of harmonic oscillators on singular geometries using supersymmetry.

Generalizing work of W. Müller we investigate the spectral theory for the Dirac operator D on a noncompact manifold X with generalized fibred cusps $C(M)=M\times [A,\infty[_r, g= d r^2+ φ^*g_Y+ e^{-2cr}g_Z,$ at infinity. Here $φ:M^{h+v}\to Y^h$ is a compact fibre bundle with fibre Z and a distinguished horizontal s…