arXiv subjects
Publications and source records attributed to Xiangyu Liang.
In this paper we discuss various minimality properties for the orthogonal product of two 1-dimensional $\Y$ sets, and some related problems. This is motivated by an attempt to give the classification of singularities for 2-dimensional Almgren-minimal sets in $\R^4$.
We discuss the global regularity for 2 dimensional minimal sets that are near a $\T$ set, that is, whether every global minimal set in $\R^n$ that looks like a $\T$ set at infinity is a $\T$ set or not. The main point is to use the topological properties of a minimal set at large scale to control its topology at smaller scales. This is the idea to prove that all 1-dimensional Almgren-minimal sets in $\R^n$, and all 2-dimensional Mumford-Shah minimal sets in $\R^3$ are cones. In this article we discuss two types of 2-dimensional minimal sets: Almgren-minimal set in $\R^3$ whose blow-in limit is a $\T$ set; topological minimal sets in $\R^4$ whose blow-in limit is a $\T$ set. For the first one we eliminate an existing potential counterexample that was proposed by several people, and show that a real counterexample should have a more complicated topological structure; for the second we construct a potential example using a Klein bottle.
In this article we prove that the union of two almost orthogonal planes in R4 is Almgren-minimal. This gives an example of a one parameter family of minimal cones, which is a phenomenon that does not exist in R3. This work is motivated by an attempt to classify the singularities of 2-dimensional Almgren-minimal sets in R4. Note that the traditional methods for proving minimality (calibrations and slicing arguments) do not apply here, we are obliged to use some more complicated arguments such as a stopping time argument, harmonic extensions, Federer-Fleming projections, etc. that are rarely used to prove minimality (they are often used to prove regularity). The regularity results for 2-dimensional Almgren minimal sets ([5],[6]) are also needed here.
In this article we introduce a definition of topological minimal sets, which is a generalization of that of Mumford-Shah-minimal sets. We prove some general properties as well as two existence theorems for topological minimal sets. As an application we prove the topological minimality of the union of two almost orthogonal planes in $\R^4$, and use it to improve the angle criterion under which the union of several higher dimensional planes is Almgren-minimal.