Locally Lipschitz contractibility of Alexandrov spaces and its applications
The authors introduce the notion of a strongly locally Lipschitz contractible metric space as follows.NEWLINENEWLINE``A metric space \(X\) is \textit{strongly locally Lipschitz contractible}, or SLLC, if for every point \(p\in X\), there exists \(r>0\) and a map NEWLINE\[NEWLINE h:U(p,r)\times[0,1]\rightarrow U(p,r) NEWLINE\]NEWLINE such that \(h\) is a homotopy from \(h(\cdot,0)=\text{id}_{U(p,r)}\) to \(h(\cdot,1)=p\) and \(h\) is Lipschitz (i.e., there exists \(C,C^\prime >0\) such that NEWLINE\[NEWLINE d(h(x,s),h(y,t))\leq Cd(x,y)+C^\prime|s-t| NEWLINE\]NEWLINE for every \(x,y\in U(p,r)\) and \(s,t\in[0,1]\)) and such that for every \(r^\prime <r\), the image of \(h\) restricted to \(U(p,r^\prime)\times[0,1]\) is \(U(p,r^\prime)\).'' Recall that the (weak) locally Lipschitz contractibility, as introduced in [the second author, Kyushu J. Math. 51, No. 2, 273--296 (1997; Zbl 0914.53028)], merely requires that for every \(p\in X\) and \(\varepsilon>0\) there exists \(r\in(0,\varepsilon]\) and a Lipschitz map \(h:U(p,r)\times[0,1]\rightarrow U(p,\varepsilon)\) such that \(h(\cdot,0)=\text{id}_{U(p,r)}\) and \(h(x,1)=p\) for every \(x\in U(p,r)\).NEWLINENEWLINENEWLINEThe main result of the authors is that every finite-dimensional Alexandrov space is strongly locally Lipschitz contractible. The proof uses the gradient flow of the distance function from a metric sphere, which function is shown to be regular on a smaller concentric punctured ball. NEWLINENEWLINENEWLINEAfter that, the authors show that a continuous map from a Lipschitz simplicial complex to a strongly locally Lipschitz contractible metric space \(X\) is homotopic to a Lipschitz one. Here, a Lipschitz simplicial complex is, roughly speaking, a metric space having a triangulation such that every simplex is a bi-Lipschitz image of a simplex in a Euclidean space. It follows that for an Alexandrov space \(X\), the \(k\)-th Lipschitz homotopy group \(\pi_k^{\text{Lip}}(X,x_0)\) and the (standard) \(k\)-th homotopy group \(\pi_k(X,x_0)\) are isomorphic. (The last two results remain true in the case of a merely locally Lipschitz contractible space \(X\), as the authors remark.) NEWLINENEWLINENEWLINEAs another application of their main result, the authors present a sufficient condition for the existence of a solution to the Plateau problem in an Alexandrov space, as introduced in [\textit{C. Mese} and \textit{P. R. Zulkowski}, J. Differ. Geom. 85, No. 2, 315--356 (2010; Zbl 1250.53066)]. NEWLINENEWLINENEWLINEFinally, the authors deduce two estimates of the so-called Gromov simplicial volume of a compact orientable \(n\)-dimensional Alexandrov space \(X\) without boundary in terms of its \(n\)-dimensional Hausdorff measure, the second estimate assuming \(X\) separable, having a locally finite Borel measure \(m\) that is absolutely continuous with respect to the \(n\)-dimensional Hausdorff measure, and such that the metric measure space \((X,m)\) satisfies the so-called reduced curvature-dimension condition locally, as introduced in [\textit{K. Bacher} and \textit{K.-T. Sturm}, J. Funct. Anal. 259, No. 1, 28--56 (2010; Zbl 1196.53027)].
scientific article; zbMATH DE number 6361409