arXiv subjects
Publications and source records attributed to Anu Dhochak.
We present a compact formula for the exact partition function of the $d$-dimensional arithmetic Ising model (AIM). For a $2\times2$ system, we express it analytically using the $q$-Hurwitz-Lerch zeta function and derive explicit forms for the free energy and entropy. Additionally, we find that the entropy increases at high temperatures, supporting the presence of entropic order.
Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $ρ:π_1(S_g)\to\PU(2,1)$, let $E_ρ$ be the energy function on Teichmüller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ ρ_{g,h,d}:π_1(S_g)\to\PU(2,1) \] with \[ τ(ρ_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{ρ_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.
In this article, we discuss the existence of a 1-parameter infinite genus family of maxfaces having infinitely many planar (spacelike) ends and infinitely many swallowtails. In particular, we show the existence of the following: (1) a period-2 family of maxfaces with infinitely many planar ends and alternating singularity types, where every odd-layer neck has exactly four swallowtails, while each even-layer neck is almost conical; for $n=2$, the fundamental piece is a genus-1 Wei-type maxface; (2) a period-3 family where every neck carries four swallowtails (24 per period); and (3) a period-2 family of maxfaces with an almost conical singularity on every neck. All maxfaces are embedded in a wider sense.
The node-opening technique, originally designed for constructing minimal surfaces, is adapted to construct a rich variety of new maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar ends, thus removing the scarcity of examples of maxfaces. The surfaces look like spacelike planes connected by small necks. Among the examples are maxfaces of the Costa--Hoffman--Meeks type. Although very fruitful, the main challenge of this paper is not the construction itself, but the analysis of the positions and natures of singularities on these maxfaces. More specifically, we conclude that the singular set form curves around the waists of the necks. In generic and some symmetric cases, all but finitely many singularities are cuspidal edges, and the non-cuspidal singularities are swallowtails evenly distributed along the singular curves.
For a given zero mean curvature surface $X$ (in the Lorentz Minkowski space) having folded singularity, we construct a family of maxface and minface, having increasing cuspidal crosscaps, converging to $X$. We include a general discussion of this.
When a connected component of the set of singular points of the maxface $X$ consists of only generalized cone-like singular points, we construct a sequence of maxfaces $X_n$, with an increasing number of swallowtails, converging to the maxface $X$. We include the general discussion toward this.