Let $(X,d,p)$ be an $n$-dimensional noncollapsed Ricci limit space, where $n\ge4$. Under an integral volume-deficit bound, we prove that the metric singular set has Hausdorff codimension at least four and sigma-finite $(n-4)$-dimensional Hausdorff measure. This establishes a special case of the codimension-four regularity conjecture. Moreover, the nonmanifold locus is closed and has locally finite $(n-4)$-dimensional Hausdorff measure, and its intersection with each bounded ball satisfies a tubular-volume estimate of order $r^4$. In dimension four, the nonmanifold points form a locally finite set. A flat quotient example shows that the codimension bound is sharp.
Let $(X,d,p)$ be a noncollapsed pointed Gromov--Hausdorff limit of complete $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where $n\ge4$. We assume that, on each bounded ball, the integral of the $3/2$ power of the small-ball volume deficit relative to the hyperbolic comparison volume...
We construct smooth collapsing sequences with sectional curvature at least $-1$ whose limits are topological manifolds with nonzero Kirby--Siebenmann class. For every $n\geq6$, the source and limit are simply connected, have dimensions $n+3$ and $n$, and the metrics have a common positive Ricci lower bound. The limits...
Given a smooth domain $D\subset\mathbb S^{N-1}$, $N\geq 3$, and a positive constant $H$, we study radial graphs over $D$ with constant mean curvature $H$ which meet orthogonally the boundary of the cone spanned by $D$. The corresponding problem is given by a quasilinear elliptic equation on $D$ with homogeneous Neumann...
We study complete two-sided constant-mean-curvature hypersurfaces of dimensions $2\le n\le6$ and finite Morse index in simply connected space forms. In the round sphere the immersed domain is compact; in Euclidean space every noncompact example is minimal; in hyperbolic space of curvature $-1$ we obtain compactness und...
We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces $A_\alpha^p(\mathbb D)$. For $p>0$ and $\alpha>-1$, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivativ...
We show that for an embedded two-sided minimal hypersurface in $\mathbb{R}^N$, there is a lower bound for the index in terms of the first Betti number and the number of ends, using ideas from the recent work of Chodosh--Gianocca. This estimate is sharp for the higher-dimensional catenoid. We also obtain a $\operatornam...
Jun-Zhang Li· 0 citations
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