The purpose of this note is to extend to an arbitrary generalized Hopf and CalabiEckmann manifold the following result of Kalyan Mukherjea: Let Vn = S 2n+1×S2n+1 denote a Calabi-Eckmann manifold. If f, g : Vn−→P are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: f(x) = g(x) for some x ∈ Vn. Our proof involves a coincidence theorem for holomor...

If G is a finitely presented group, then H2 (∞) (G,A) vanishes for all injective Banach spaces iff the regularized homological area function satisfies the linear isoperimetric inequality. This contrasts with the known result that G is word hyperbolic iff the homological area function satisfies the linear isoperimetric inequality. A closed 3-manifold group G is hyperbolic iff H2 (∞) (G,A) vanish...

For every finitely generated free group $F$, we construct an irreducible open $3$-manifold $M_F$ whose end set is homeomorphic to a Cantor set, and with the homogeneity of isomorphic $F$. The all self-homeomorphisms that extend homeomorphisms entire $3$-manifold. This extends earlier result constructs, for each abelian $G$, $M_G$ $G$. method used in proof our main also shows if $G$ Cayley graph...

Let (M, g) be a closed Riemannian manifold of dimension n ≥ 3 and let f ∈ C∞(M), such that the operator Pf := ∆g + f is positive. If g is flat near some point p and f vanishes around p, we can define the mass of Pf as the constant term in the expansion of the Green function of Pf at p. In this paper, we establish many results on the mass of such operators. In particular, if f := n−2 4(n−1) sg, ...

The family P of all convex 3-polytopes P in Euclidean space E 3 may be partitioned into combinatorial types or configuration spaces by isomorphism of face lattices , and the configuration space [ ] P of any such 3-polytope P may be subdivided further into GConfiguration space P by equivalence of actions of symmetry group ) (P G on face lattices. With respect to a natural topology induced by Hau...

The aim of the present paper is to study a Bochner Ricci semi-symmetric quasi-Einstein Hermitian manifold (QEH)n, a Bochner Ricci semi-symmetric generalised quasi-Einstein Hermitian manifold G(QEH)n and a Bochner Ricci semisymmetric pseudo generalised quasi-Einstein Hermitian manifold P (GQEH)n.

∂tgij = −2Rij , on MT := M × [0, T ) where Rij is the Ricci tensor of the metric g := g(t) and T is the maximal existing time for the flow. In [2], R.Hamilton proved the local existence of the flow for the compact manifold case. His argument is much simplified by De Turck [1]. When (M,g0) is a a complete non-compact Riemannian manifold with bounded geometry, W.X.Shi [5] obtained the local exist...

We develop geometric optimisation on the manifold of hermitian positive definite (hpd) matrices. In particular, we consider optimising two types of cost functions: (i) geodesically convex (g-convex); and (ii) log-nonexpansive (LN). G-convex functions are nonconvex in the usual euclidean sense, but convex along the manifold and thus allow global optimisation. LN functions may fail to be even g-c...

We show that 3-MANIFOLD KNOT GENUS, the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most g, is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NP-hard.