scholarly journals Heat asymptotics with spectral boundary conditions. II

2003 ◽  
Vol 133 (2) ◽  
pp. 333-361 ◽  
Author(s):  
Peter Gilkey ◽  
Klaus Kirsten
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Dirac Type ◽  
Heat Trace ◽  
Operator Of Dirac Type ◽  
Laplace Type

Let P be an operator of Dirac type on a compact Riemannian manifold with smooth boundary. We impose spectral boundary conditions and study the asymptotics of the heat trace of the associated operator of Laplace type.

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

scholarly journals Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary

2017 ◽  
Vol 8 (1) ◽  
pp. 559-582 ◽  
Author(s):  
Mónica Clapp ◽  
Marco Ghimenti ◽  
Anna Maria Micheletti
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Compact Riemannian Manifold ◽  
Semiclassical Limit ◽  
Arbitrary Dimension ◽  
Neumann Boundary ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Manifold With Boundary ◽  
Klein Gordon

Abstract We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold {\mathfrak{M}} with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of {\mathfrak{M}} , forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in {\mathbb{R}^{N}} . Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

scholarly journals A heat trace anomaly on polygons

2015 ◽  
Vol 159 (2) ◽  
pp. 303-319 ◽  
Author(s):  
RAFE MAZZEO ◽  
JULIE ROWLETT
Keyword(s):  
Boundary Conditions ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Higher Dimensions ◽  
Compact Surface ◽  
Neumann Boundary Conditions ◽  
Piecewise Smooth Boundary ◽  
Trace Anomaly ◽  
Heat Trace ◽  
Heat Invariants

AbstractLet Ω0be a polygon in$\mathbb{R}$2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωεis a family of surfaces with${\mathcal C}$∞boundary which converges to Ω0smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient oft0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domainZwhich models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

2017 ◽  
Vol 59 (3) ◽  
pp. 743-751
Author(s):  
SHOUWEN FANG ◽  
FEI YANG ◽  
PENG ZHU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Normalized Ricci Flow ◽  
Geometric Operator

AbstractLet (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

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