Title
The large scale structure of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
Talk
During the talk we will consider smooth, complete four manifolds with nonnegative Ricci curvature. If we have a ball and increase its radius the volume should grow quicker than for some .
Our goal today is to show that the manifold looks like (isometric) a cone . The section of the cone is homeomorphic to a spherical space form (i.e. a sphere modulo a form actin on it). (imagine the section being the base)
Background
Gromov 1981 showed that given a sequence of manifolds with Riemannian metric whose curvature is greater than some value converge to a pointed metric space in the pointed Gromov-Hausdorff sense.
Lets fix a smooth complete Riemannsche Mannigfaltigkeit with nonnegative Ricci-curvature and we take a real sequence. We then apply Gromov’s theorem. When scaling the metric by the inverse of the real numbers, the curvature will get smaller and in the limit it will converge to something called a blowdown . We can imagine how the hyperboloid gets blown down to a cone in the limit case (i.e. the curvature goes to zero)
Chengen-Coldung 1996 showed if the volume grows euclidianly, then the limit is a matric cone over the metric cone (i.e. the cone base)
Sturm, Lott-Villani, Ambrosio-Gigle-Savone showed that every cone obtained this way has not r a reality.
Kettenen (113) also schowed something but I didnnd “get itl”
Hamiton 1982 appliesa and it tells us, that the base is momeomoru used the properties described above to constrcut
Main result
The main result was by Mrie, Pigeti and Semola. They showed there is a finifte acting freely and
Optimality of theorem
- Perelmann 1993 konstruierte Beipiele mit nicht-eindeutigem blow-down.
- Zhou zeigte dass jeder Quitient der Kugel kann nur im unendlichen Geschehen. Ebenfalls beteiligt daran waren eine Menge anderer Menschen.
- In höherendimensionalen Fällen is die Basis nicht notwengerweise glatt sondern nur eine Topologische Mannigfaltigkeit.
- Coldin-Noben have constructed exmaples of dimension greater than five. with two distinct blow-downs..
Earlier developments
- Im Falle ist der Blowdow eindeutig und von einer speziellen Form.
- Im Falle könnte der Blowdown schon nicht mehr eindeutig sein. But they are of a specific form.
(There is a lot of mathematics by Gromov and Perelman. This is really scary stuff) Afterwards we present a lot of new conditions and resulting results.
Next we will present some elements of the proof (I think I’m going to write my diary now)
Wait, why is there a new Background? In the opposite way we constructed the blow-down, we can construct a blow up of a metric space by scaling up the metric. If this converges to the Gromov-Hausdorff-something, the result is actually called a blow-up.
The existance of a blowup is not always there. For some spaces they are guaranteed to exist though. If the blwoup is of a given form we deduce even more properties.
Open Questions
- What can we say about the topology of if is contractible
- Another question