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Register a new userAdditive higher rho invariant for structure group in differential point of view
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In this paper, we adapt part of Weinberger, Xie and Yus breakthrough work, to define additive higher rho invariant for topological structure group by differential geometric version of signature operators, or in other words, unbounded Hilbert-Poincar{e} complexes.
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In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a higher index theorem for the Dirac operators. We apply our theory to study the secondary invariants for a manifold with corner with positive scalar curvature metric on each boundary face.
Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Omega$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of total Gauss curvature of a conformal metric on $Omega$. Consequently, new analytic and geometric isoperimetric-type inequalities are discovered. Furthermore, when applying the geometric principle to two-dimensional Riemannian manifolds, we find fortunately that ${0,1}$-form of the induced principle is midway between Moser-Trudingers inequality and Nash-Sobolevs inequality on complete noncompact boundary-free surfaces, and yet equivalent to Nash-Sobolevs/Faber-Krahns eigenvalue/Heat-kernel-upper-bound/Log-Sobolevs inequality on the surfaces with finite total Gauss curvature and quadratic area growth.
Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for higher index and higher rho invariant of signature operator on fibered manifolds. Our result implies the classical product formula for numerical signature of fiber manifolds obtained by Chern, Hirzebruch, and Serre in On the index of a fibered manifold. We also give a new proof of the product formula for higher rho invariant of signature operator on product manifolds, which is parallel to the product formula for higher rho invariant of Dirac operator on product manifolds obtained by Xie and Yu in Positive scalar curvature, higher rho invariants and localization algebras and Zeidler in Positive scalar curvature and product formulas for secondary index invariants.
In this paper we consider successive iterations of the first-order differential operations in space ${bf R}^3.$
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
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