On a smooth closed oriented 4-manifold $$M$$
 with a smooth action by a finite group $$G$$
 , we show that a $$G$$
 -monopole class gives the $$L^2$$
 -estimate of the Ricci curvature of a $$G$$
 -invariant Riemannian metric, and derive a topological obstruction to the existence of a $$G$$
 -invariant nonsingular solution to the normalized Ricci flow on $$M$$
 . In particular, for certain $$m$$
 and $$n, m\mathbb C P_2 \# n\overline{\mathbb{C P}}_2$$
 admits an infinite family of topologically equivalent but smoothly distinct non-free actions of $$\mathbb Z _d$$
 such that it admits no nonsingular solution to the normalized Ricci flow for any initial metric invariant under such an action, where $$d>1$$
 is a non-prime integer. We also compute the $$G$$
 -Yamabe invariants of some 4-manifolds with $$G$$
 -monopole classes and the orbifold Yamabe invariants of some 4-orbifolds.

G-monopole classes, Ricci flow, and Yamabe invariants of 4-manifolds

On a smooth closed oriented 4-manifold $M$ with a smooth action by a finite group $G$, we show that a $G$-monopole class gives the $L^2$-estimate of the Ricci curvature of a $G$-invariant Riemannian metric, and derive a topological obstruction to the existence of a $G$-invariant nonsingular solution to the normalized Ricci flow on $M$. 
In particular, for certain $m$ and $n$, $m\Bbb CP_2 # n\bar{\Bbb CP}_2$ admits an infinite family of topologically equivalent but smoothly distinct non-free actions of $\Bbb Z_d$ such that it admits no nonsingular solution to the normalized Ricci flow for any initial metric invariant under such an action, where $d>1$ is a non-prime integer. 
We also compute the $G$-Yamabe invariants of some 4-manifolds with $G$-monopole classes and the oribifold Yamabe invariants of some 4-orbifolds.

The Yamabe invariant is an invariant of a closed smooth manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold T^m\times B where T^m$ is the m-dimensional torus, and B is a closed spin manifold of nonzero \hat{A}-genus has zero Yamabe invariant. We generalize it to various T-structured manifolds, for example T^m-bundles over such B whose transition functions take values in Sp(m,Z) (or Sp(m-1,Z)\oplus \pm 1 for odd m).

T-structure and the Yamabe invariant

. We present smooth simply connected closed 4 k -dimensional manifolds N := N k , for each k 2 f 2 ; 3 ;¢¢¢g , with distinct symplectic deformation equivalenceclasses [[ ! i ]], i = 1 ; 2. To distinguish [[ ! i ]]’s, we used the symplectic Z invariant in[4] which depends only on the symplectic deformation equivalence class. We havecomputed that Z ( N; [[ ! 1 ]]) = 1 and Z ( N; [[ ! 2 ]]) < 0. 1. Introduction An almost-K˜ahler metric on a smooth manifold M 2 n of real dimension 2 n is aRiemannian metric g compatible with a symplectic structure ! , i.e. ! ( X;Y ) = g ( X;JY ) for an almost complex structure J , where X;Y are tangent vectors at apoint of the manifold. Two symplectic forms ! 0 and ! 1 on M are called deformationequivalent , if there exists a diﬁeomorphism ˆ of M such that ˆ ⁄ ! 1 and ! 0 can bejoined by a smooth homotopy of sympelctic forms, [5]. For a symplectic form ! , itsdeformation equivalence class shall be denoted by [[ ! ]]. We denote by › [[ ! ]] the setof all almost K˜ahler metrics compatible with a symplectic form in [[

SIMPLY CONNECTED MANIFOLDS OF DIMENSION 4k WITH TWO SYMPLECTIC DEFORMATION EQUIVALENCE CLASSES

Let M be a smooth closed manifold of dimension 4 k + k ′ (possibly with orbifold singularities) in which a smooth closed manifold F of dimension k ′ is embedded with trivial normal bundle, and H P k × ˜ F be a smooth fiber bundle over H P k with fiber F. Denoting the Yamabe invariant of a manifold X by Y ( X ) , we show that if Y ( M ) ≤ 0 ≤ Y ( H P k × ˜ F ) and | Y ( H P k × ˜ F ) | + | Y ( M ) | ≠ 0 , then Y ( M ) = Y ( M ♯ F ( H P k × ˜ F ) ) , where ♯ F denotes a generalized connected sum along F. Likewise for the Cayley plane C a P 2 when k = 4 . These results are used to produce examples of higher-dimensional manifolds with nonpositive Yamabe invariants.

Connected sums with HPn or CaP2 and the Yamabe invariant

For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kahler metrics, i.e., Riemannian metrics compatible with symplectic structures. We also introduce $$Z(M, [[\omega ]])$$
 depending on symplectic deformation equivalence class $$[[\omega ]]$$
 . We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of $$Z(M, [[\omega ]] )$$
 . Using Z invariants, we set up a Kazdan–Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold $$(M, \omega )$$
 of dimension $$\ge \!\!4$$
 , any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kahler metric compatible with a symplectic form which is deformation equivalent to $$\omega $$
 .

Scalar Curvature Functions of Almost-Kähler Metrics

We show that the normalized Ricci flow $g(t)$ on a smooth closed manifold $M$ existing for all $t \geq 0$ with scalar curvature converging to constant in $L^2$ norm should satisfy $$\liminf_{t\rightarrow \infty} \int_M|\stackrel{\circ}{r}|_{g(t)}^2d\mu_{g(t)} =0,$$ where $\stackrel{\circ}{r}$ is the trace-free part of Ricci tensor. Using this, we give topological obstructions to the existence of such a Ricci flow (even with positive scalar curvature tending to $\infty$ in sup norm) on 4-manifolds.

On the normalized Ricci flow with scalar curvature converging to constant

We further sharpen higher type adjunction inequalities of Ozsvath and Szabo on a 4-manifold M with a non-zero Seiberg–Witten invariant for a Spinc structure &#x1d530;, when an embedded surface Σ ⊂ M satisfies [Σ] ⋅ [Σ] ≥ 0 and

Some refined higher type adjunction inequalities on 4-manifolds

On a smooth closed oriented $4$-manifold $M$ with a smooth action of a finite group $G$ on a Spin$^c$ structure, $G$-monopole invariant is defined by "counting" $G$-invariant solutions of Seiberg-Witten equations for any $G$-invariant Riemannian metric on $M$. 
We compute $G$-monopole invariants on some $G$-manifolds. For example, the connected sum of $k$ copies of a 4-manifold with nontrivial mod 2 Seiberg-Witten invariant has nonzero $\Bbb Z_k$-monopole invariant mod 2, where the $\Bbb Z_k$-action is given by cyclic permutations of $k$ summands.

Chanyoung Sung

Papers

Connected sums with HPn or CaP2 and the Yamabe invariant

Scalar Curvature Functions of Almost-Kähler Metrics

On the normalized Ricci flow with scalar curvature converging to constant

Some refined higher type adjunction inequalities on 4-manifolds

G-monopole invariants on some connected sums of 4-manifolds