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Authors
Richard H. Bamler
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In this paper we generalize the theory of Cheeger, Colding and Naber to certain singular spaces that arise as limits of sequences of Riemannian manifolds. This theory will have applications in the analysis of Ricci flows of bounded curvature, which we will describe in a subsequent paper.
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In this paper, we will show the Yaus gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $kappa$, $kappageq0$, in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.
In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension $geq 4$ in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension $geq 4$. We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds. As an application of our theory, we obtain a description of the singularity formation of a Ricci flow at its first singular time and a thick-thin decomposition characterizing the long-time behavior of immortal flows. These results generalize Perelmans results in dimension 3 to higher dimensions. We also obtain a Backwards Pseudolocality Theorem and discuss several other applications.
We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatchers theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case.
By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=frac{da^2}{h(a^2)}+a^2g_{S^n}$ for some function $h$ where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $mathbb{R}^n$ for any $nge 2$. More precisely for any $lambdage 0$ and $c_0>0$, we prove that there exist infinitely many solutions $hin C^2((0,infty);mathbb{R}^+)$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-lambda r-(n-1))$, $h(r)>0$, in $(0,infty)$ satisfying $underset{substack{rto 0}}{lim},r^{sqrt{n}-1}h(r)=c_0$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin.
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Holder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Holder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.
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