## Asian Journal of Mathematics

- Asian J. Math.
- Volume 18, Number 4 (2014), 609-622.

### Small four-manifolds without non-singular solutions of normalized Ricci flows

#### Abstract

It is known that connected sums $X\# K 3 \# (\Sigma_g \times \Sigma_h) \# \ell_1 (S^1 \times S^3) \# \ell_2 \overline{\mathbb{C}P^2}$ satisfy the Gromov-Hitchin-Thorpe type inequality, but can not admit non-singular solutions of the normalized Ricci flow for any initial metric, where $\Sigma_g \times \Sigma_h$ is the product of two Riemann surfaces of odd genus, $\ell_1, \ell_2 \gt 0$ are sufficiently large positive integers, $g, h \gt 3$ are also sufficiently large positive odd integers, and $X$ is a certain irreducible symplectic 4-manifold. These examples are closely related with a conjecture of Fang, Zhang and Zhang. In the current article, we point out that there still exist 4-manifolds with the same property even if $\ell_1 = \ell_2 = 0$ and $g = h = 3$. The topology of these new examples are smaller than that of previously known examples.

#### Article information

**Source**

Asian J. Math., Volume 18, Number 4 (2014), 609-622.

**Dates**

First available in Project Euclid: 6 November 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1415284980

**Mathematical Reviews number (MathSciNet)**

MR3275721

**Zentralblatt MATH identifier**

1308.53100

**Subjects**

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX] 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

**Keywords**

Four-manifold Ricci flow non-singular solution

#### Citation

Ishida, Masashi. Small four-manifolds without non-singular solutions of normalized Ricci flows. Asian J. Math. 18 (2014), no. 4, 609--622. https://projecteuclid.org/euclid.ajm/1415284980