arXiv subjects

Kazuo Akutagawa

Publications and source records attributed to Kazuo Akutagawa.

22 records · Page 2Linked to original sources

arXiv2003

Yamabe metrics on cylindrical manifolds

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding version of the Yamabe problem on cylindrical manifolds. We affirmatively solve this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almost cusp singularities. We describe the supremum case, i.e. when the cylindrical Yamabe constant is equal to the Yamabe invariant of the sphere. We prove that in this case such a cylindrical manifold coincides conformally with the standard sphere punctured at a finite number of points. In the course of studying the supremum case, we establish a Positive Mass Theorem for specific asymptotically flat manifolds with two almost conical singularities. As a by-product, we revisit known results on surgery and the Yamabe invariant. Key words: manifolds with cylindrical ends, Yamabe constant/invariant, Yamabe problem, conical metric singularities, cusp metric singularities, Positive Mass Theorem, surgery and Yamabe invariant.

math.DG↗

arXiv2002

The Weyl functional near the Yamabe invariant

For a compact manifold $M$ of $\dim M =n\geq 4$, we study two conformal invariants of a conformal class $C$ on $M$. These are the Yamabe constant $Y_C(M)$ and the $L^{\frac{n}{2}}$-norm $W_C(M)$ of the Weyl curvature. We prove that for any manifold $M$ there exists a conformal class $C$ such that the Yamabe constant $Y_C(M)$ is arbitrarily close to the Yamabe invariant $Y(M)$, and, at the same time, the constant $W_C(M)$ is arbitrarily large. We study the image of the map $\YW: C\mapsto (Y_C(M),W_C(M))\in \R^2$ near the line $\{(Y(M),w) | w\in \R\}$. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.

math.DG↗

arXiv2000

Relative Yamabe Invariant

We define a relative Yamabe invariant of a smooth manifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the classic Yamabe invariant. We develop approximation technique which leads to gluing theorems of two manifolds along their boundaries for the relative Yamabe invariant. We show that there are many examples of manifolds with both positive and non-positive relative Yamabe invariants.

math.DG↗

arXiv2000

Manifolds of Positive Scalar Curvature and Conformal Cobordism Theory

We study here compact manifolds with positive scalar curvature metrics. We use the relative Yamabe invariant from math.DG/0008138 to define the conformal cobordism relation on the category of such manifolds. We prove that corresponding conformal cobordism groups $\Pos_n^{\conf}(γ)$ are isomorphic to the cobordism groups $\Pos_n(γ)$ defined topologically by S. Stolz. As a corollary we show that the conformal concordance of positive scalar curvature metrics coincides with the standard concordance relation. Our main technical tools came from the analysis and conformal geometry.

math.DG↗