A matrix Harnack inequality for semilinear heat equations

<abstract><p>We derive a matrix version of Li &amp; Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in ^{[<xref ref-type="bibr" rid="b5">5</xref>]} for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

On the geometry of the tangent bundle with gradient Sasaki metric

PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

Nilpotent Higgs Bundles and the Hodge Metric on the Calabi–Yau Moduli

We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi–Yau moduli.

Almost isotropic Kähler manifolds

Let M be a complete Riemannian manifold and suppose {p\in M}. For each unit vector {v\in T_{p}M}, the Jacobi operator, {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v}. Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M}. If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds, i.e. manifolds having the property that for each {p\in M}, there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M}. Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}}.

Four-Dimensional Almost Einstein Manifolds with Skew-Circulant Structres

We consider a four-dimensional Riemannian manifold M with an additional structure S, whose fourth power is minus identity. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of Einstein and almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some characteristic 2-planes in a tangent space of M. We consider a Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a K\"{a}hler manifold. We construct some examples of the considered manifolds on Lie groups.

FOUR-MANIFOLDS WITH POSITIVE CURVATURE

In this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M 4 is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$ , provided that the sectional curvatures all lie in the interval $\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

On the Betti and Tachibana Numbers of Compact Einstein Manifolds

Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold ( M , g ) with an Einstein constant α > 0 is a homological sphere when the minimum of its sectional curvatures > α / ( n + 2 ) ; in particular, ( M , g ) is a spherical space form when the minimum of its sectional curvatures > α / n . Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold with α < 0 .

Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $${{\,\mathrm{\mathbb {R}}\,}}^n$$Rn with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $$ M^n \times {{\,\mathrm{\mathbb {R}}\,}}$$Mn×R. Precisely, given a $$\mathscr {C}^2$$C2 bounded domain $$\Omega $$Ω in M and a function $$ H = H (x, z) $$H=H(x,z) continuous in $$\overline{\Omega }\times {{\,\mathrm{\mathbb {R}}\,}}$$Ω¯×R and non-decreasing in the variable z, we prove that the strong Serrin condition$$(n-1)\mathcal {H}_{\partial \Omega }(y)\ge n\sup \limits _{z\in {{\,\mathrm{\mathbb {R}}\,}}}\left| H(y,z) \right| \ \forall \ y\in \partial \Omega $$(n-1)H∂Ω(y)≥nsupz∈RH(y,z)∀y∈∂Ω, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins–Serrin and Serrin type sharp solvability criteria.