arXiv research

Khovanov homology invariant proved for links in $\mathbb{RP}^3$.

We prove the $3$-manifold $\RP^3 \# \RP^3$ is of $\Z_{2}$-coefficient homology $(1, 2)$-systolic freedom. Given a Riemannian metric on $\RP^{3}\# \RP^{3}$, we define $\Z_{2}$-coefficient homology $1$-systole as the infimum of lengths of all nonseparating geodesic loops representing nontrivial classes in $H_{1}(\RP^3\#\…

Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal $\mathbb{Z}_2^2$-equivariant triangulations of $2$-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{R…

P. Arnoux and A. Marin showed that any triangulation of $\mathbb{RP}^n$ contains more than $\frac{(n+1)(n+2)}{2}$ vertices if $n \geq 3$. We construct some natural triangulation of $\mathbb{RP}^n$ with $\frac{n(n+5)}{2}-1$ vertices for all $n \geq 3$. Previously, it was known that $\mathbb{RP}^n$ has $\mathbb{Z}_2^n$-e…

New proof of Yamabe invariant for RP^3 using harmonic functions.

We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgal…

Study instanton Floer homology for links in RP^3 and use it to detect knots.

Paper introduces Bar-Natan homology for special links in a modified space.

New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.

Defines a Rasmussen invariant for links in RP^3.

The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.

In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms $f:M \to M$ for a certain class of PL-manifolds $M$. These yield upper bounds for gem-complexity and regular genus of a large class of PL-manifolds. The bound for the regular genus is sharp for the mapping torus of some (PL…

New homologies defined for null homologous links in RP^3, linking to Heegaard Floer homology.

Links with isotopic preimages in $S^3$ are isotopic in $RP^3$.

The affine sphere construction gives, on any oriented surface, a one-to-one correspondence between convex $\mathbb{RP}^2$-structures and holomorphic cubic differentials. Generalizing results of Benoist-Hulin, Loftin and Dumas-Wolf, we show that poles of order less than $3$ of cubic differentials correspond to finite vo…

The study constructs metrics with positive 2nd Ricci curvature on various manifolds.

Fixed points found in Teichmüller space via anti-de Sitter geometry.

Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.

New link homology theories for $\mathbb{RP}^3$ yield distinct invariants.

New proof for 3-manifolds with specific homology groups.

Study on projective structures and their foliations on surfaces.

The regular genus of certain 4-manifolds is determined, providing new insights.

The study proves representations for certain 3-manifolds using gauge theory.

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $Σ\times \RP$ with Nahm pole singularity at $Σ\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop…

New bounds on slice genus from knot invariants.

Study real projective structures on a specific Coxeter orbifold.

We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics…

Any compact manifold with positive scalar curvature has an associated asymptotically flat metric constructed using the Green's function of the conformal Laplacian, and the mass of this metric is an important geometric invariant. An explicit expression for the mass of the product of spheres $S^2 \times S^2$, both with t…

We address a long-standing and long-investigated problem in combinatorial topology, and break the exponential barrier for triangulations of real projective space, constructing a trianglation of $\mathbb{RP}^n$ of size $e^{(\frac{1}{2}+o(1))\sqrt{n}{\log n}}$.

New method uses Floer homology to create exotic disk pairs.

The study connects polygon areas and projective structures in 3D space.

Let (M,g) be a compact Riemannian manifold of dimension 3, and let \mathscr{F} denote the collection of all embedded surfaces homeomorphic to \mathbb{RP}^2. We study the infimum of the areas of all surfaces in \mathscr{F}. This quantity is related to the systole of (M,g). It makes sense whenever \mathscr{F} is non-empt…

Minimal surfaces in lens spaces identified with specific counts.

Enhanced Khovanov TQFT using basepoints for nonorientable surfaces.

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with the conformal extremal metric on the regular $2n$-gon. The hexagon was considered…

Study bounds the volume of moduli space for convex RP² structures.

The article studies crystallizations of small covers over simple polytopes and finds unique crystallizations for the $n$-simplex.

The universe's shape and size are determined in general cosmological models.

For a given real generic curve $\ga: S^1\to \Bbb {RP}^n$ let $D_\ga$ denote the ruled hypersurface in $\Bbb {RP}^n$ consisting of all osculating subspaces to $\ga$ of codimension 2. A curve $\ga: S^1\to \Bbb {RP}^n$ is called convex if the total number of its intersection points (counted with multiplicities) with any h…

We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces wit…

We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake RP^{6}s of Hirsch and Milnor and on the analogous fake RP^{14}s.

We construct integrable hierarchies of flows for curves in centroaffine ${\mathbb R}^3$ through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …

Researchers compute Floer homotopy types and eta invariants for Seifert 3-manifolds.

In this note we prove the following result: Let $X$ be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry and with no essential incompressible space form. Then $X$ is diffeomorphic to $\mathbb{S}^4$, or $\mathbb{RP}^4$, or $\mathbb{S}^3\times \mathbb{S}^1$, or $\mathbb{S…

In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.

Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$, the complete graph on ten vertices, contains a non-split three-component link; that is, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}^3$. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrin…

We show that the manifold *CP^2 # *RP^4, which is homotopy equivalent but not homeomorphic to CP^2 # RP^4, is in fact smoothable.

We study minimal harmonic maps $g: {\mathbb{C}} \to SO(3) \backslash SL(3,{\mathbb{R}})$, parameterized by polynomial cubic differentials $P$ in the plane. The asymptotic structure of such a $g$ is determined by a convex polygon $Y(P)$ in ${\mathbb{RP}^2}$. We give a conjectural method for determining $Y(P)$ by solving…