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Register a new userRemarks on Nahms equations
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Authors
Nigel Hitchin
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Nahms equations are viewed in a more general context where they appear as a vector field on a moduli space of co-Higgs bundles on the projective line. Zeros of this vector field correspond to torsion-free sheaves on a singular spectral curve which we translate in terms of a smooth curve in three-dimensional projective space. We also show how generalizations of Nahms equations are required when the spectral curve is non-reduced and deduce the existence of non-classical conserved quantities in this situation.
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The moduli space of solutions to Nahms equations of rank (k,k+j) on the circle, and hence, of SU(2) calorons of charge (k,j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on P^1xP^1 trivialized at infinity with c_2=k and equipped with a flag of degree j along P^1x{0}. An explicit matrix description of these spaces is given by a monad construction
We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Greens function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n geq 10$, let $(M^n,g)$ be a closed Einstein manifold of positive scalar curvature and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a maximum point of $f$ and $f$ satisfies a vanishing order condition at this maximum point, then there exists a conformal metric $tilde g$ of $g$ such that its $Q$-curvature $Q_{tilde g}^6$ equals $f$.
We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
In this paper, we prove that a compact quasi-Einstein manifold $(M^n,,g,,u)$ of dimension $ngeq 4$ with boundary $partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere $Bbb{S}^n_+,$ or $g=dt^{2}+psi ^{2}(t)g_{L}$ and $u=u(t),$ where $g_{L}$ is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension $n=3$ is also discussed.
In this paper, we prove that the class of bi-f-harmonic maps and that of f-biharmonic maps from a conformal manifold of dimension not equal to 2 are the same (Theorem 1.1). We also give several results on nonexistence of proper bi-f-harmonic maps and f-biharmonic maps from complete Riemannian manifolds into nonpositively curved Riemannian manifolds. These include: any bi-f-harmonic map from a compact manifold into a non-positively curved manifold is f-harmonic (Theorem 1.6), and any f-biharmonic (respectively, bi-f-harmonic) map with bounded f and bounded f-bienrgy (respectively, bi-f-energy) from a complete Riemannian manifold into a manifold of strictly negative curvature has rank < 2 everywhere (Theorems 2.2 and 2.3).
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