Uniqueness of regular tangent cones for immersed stable hypersurfaces (Q6651306)

Let \(M^{n}\) be a minimal hypersurface in an Euclidean space with \(0\in sing(M)= \overline{M}\setminus M\) a singular point of \(M\). The monotonicity of the mass ratio \[r\rightarrow r^{-n}\mathcal{H}^{n}(M\cap B_{r}(0))\] implies that for any sequence of radii \(r_{i}\downarrow 0\), there is a subsequence \(r_{i^{\prime}}\) such that, as varifolds, \(r_{i^{\prime}}^{-1}M\rightharpoonup \mathbf{C}\), a limit varifold called tangent cone to \(M\) at \(0\), because of the monotonicity formula for the mass ratio. If the mass ratio is bounded at infinity, i.e., \[\limsup_{r\rightarrow \infty}r^{-n}\mathcal{H}^{n}(M\cap B_{r}(0))<\infty,\] \Nthen taking a sequence of radii \(r_{i}\uparrow \infty\), the mass ratio upper bound allows extracting limits \[r_{i^{\prime}}^{-1}M\rightharpoonup \mathbf{C},\] \Ncalled tangent cone at infinity or blow-down cone. The purpose of this paper is to derive asymptotic properties of \(M\) from \(\mathbf{C}\), in the case of stable immersed minimal hypersurfaces (with a small singular set), when \(\mathbf{C}\) is a regular cone, i.e., \(sing(\mathbf{C})=\{0\}\). \N\NThe main result considers an immersed stable minimal hypersurface \(M\) in \(B_{1}^{n+1}(0)\subset \mathbb{R}^{n+1}\) (resp.\(\mathbb{R}^{n+1}\setminus B_{1}^{n+1}(0)\)), with \[\mathcal{H}^{n-2}(sing(M))=0.\] If there is a sequence \(r_{i}\rightarrow 0\) (resp.\ \(r_{i}\rightarrow \infty\)) such that \[r_{i}^{-1}M\rightarrow q\left\vert \mathbf{C}\right\vert\] as varifolds, where \(C\) is a regular cone, then, \(r^{-1}M\rightarrow q\left\vert \mathbf{C}\right\vert\) as \(r\rightarrow 0\) (resp. \(r\rightarrow \infty\)), and the convergence rate is bounded by \(C\left\vert log(r)\right\vert ^{-\gamma}\), for some constants \(C\) and \(\gamma >0\) depending only on \(\mathbf{C},q\). More precisely, there is a (smooth) \(q\)-valued function \[u:\mathbf{C}\cap B_{1/2}^{n+1}(0)\rightarrow \mathcal{A}_{q}(\mathbf{C}^{\perp})\] (resp. \(u: \mathbf{C}\setminus \overline{B}_{2}^{n+1}(0)\rightarrow \mathcal{A}_{q}(\mathbf{C}^{\perp})\)) with \(\text{graph}(u)=M\), \(u\) having the claimed decay rate at \(0\) (resp. \(\infty\)). Here \(\mathcal{A}_{q}(\mathbb{R})\) is the set of unordered \(q\)-tuples of real numbers. \N\NThe authors prove a quite similar result in the case where \(M\) is a stationary integral varifold in \(B_{1}^{n+1}(0)\subset \mathbb{R}^{n+1}\) (resp. \(\mathbb{R}^{n+1}\setminus B_{1}^{n+1}(0)\)) which has a stable regular part and no triple junction singularities. They finally consider the case where \(\mathbf{C}\) is a regular hypercone such that its link is not simply connected. For the proof, the authors adapt the Łojasiewicz-Simon-type decay estimates to the immersed setting, using the sheeting theorem for stable minimal immersions due to \textit{C. Bellettini} [``Extensions of Schoen--Simon--Yau and Schoen--Simon theorems via iteration à la De Giorgi'', Preprint, \url{arXiv:2310.01340}]. Because covers do not always respect the cylindrical structure, they directly prove the decay at the level of \(q\)-valued graphs, and they obtain a Dini-type a priori estimate.