arXiv subjects

Santiago R. Simanca

Publications and source records attributed to Santiago R. Simanca.

17 recordsLinked to original sources

arXiv2026

The sigma invariant of the $n$ torus, the K3 surface, and Euclidean and elliptic 3d manifolds

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature$Θ_{f_g}(M)$, and squared $L^2$ norm of the mean curvature vector $Φ_{f_g}(M)$ and second fundamental form $Π_{f_g}(M)$ functionals of $f_g$, respectively. Then $\mc{W}_{f_g}(M) =(1-δ_{n,1})(n/(n-1))Θ_{f_{g}}(M) + Φ_{f_{g)}}(M)$ and $\mc{D}_{f_g}(M)=(1- δ_{n,1}) (1/(n-1)) Θ_{f_g}(M)+Π_{f_{g)}}(M)$ are functionals intrinsically defined in the space of metric in the conformal class of $g$, and $\mc{S}_g(M):=\int s_g dμ_g=\mc{W}_{f_g}(M)- \mc{D}_{f_g}(M)$. We extend the notions of $σ$ invariant and Kazdan-Warner type to manifolds of dimension $n\geq 1$. $M$ is a manifold of type II if, and only if, it admits a Ricci flat metric $g$ with minimal isometric embedding $f_g$ that minimizes $\mc{W}_{f_{g'}}(M)$ and $\mc{D}_{f_{g'}}(M)$ among metrics $g'$ in conformal classes $[g']$ with scalar flat representatives. We show that the torus $T^n$, the K3 surface, and any Euclidean 3d manifold are manifolds of Kazdan-Warner type II, exhibiting in each case the canonical Ricci flat $g$ that realizes the vanishing $σ$ invariant and said minimal value $\mc{W}_{f_g}(M)= \mc{D}_{f_g}(M)$, with Euclidean 3d manifolds of isomorphic $π_1$ being diffeomorphic iff the values of $\mc{W}_{f_g}(M)$ for their canonical $g$s are the same. An elliptic 3d manifold $(M,Γ_M)$ of underlying group $π_1(M)\cong Γ_M \subset \mb{S}\mb{O}(4)$ has $σ(M) =6(2π^2)^{ \frac{2}{3}}/|π_1(M)|^{\frac{2}{3}}$, and if $(M,Γ_M)$ and $(M',Γ_{M'})$ are two of them of isomorphic $π_1$, $M$ is diffeomorphic to $M'$ iff the spaces of $Γ_M$ and $Γ_{M'}$ invariant homogeneous spherical harmonics of degree $|π_1|$ are the same.

math.DG↗

arXiv2026

Minimally embedded Riemann surfaces in $\mb{S}^3$ and the conformal deformation of their metrics

We prove that if $f_g: (Σ,g) \rightarrow (\mb{S}^{2+p},\tg)$ is a smooth minimal isometric embedding of a Riemannian surface $(Σ,g)$, and $[0,1]\ni t \rightarrow g_t$ is a path of area preserving conformal deformations of $g$ on $f_g(Σ)$, then there exists a path of conformal diffeomorphism $F_t: (\mb{S}^{2+p}, F_t^*\tg) \rightarrow (\mb{S}^{2+p},\tg)$ that starts at $\BOne_{\mb{S}^{2+p}}$, set theoretically fixes $f_g(Σ)$ for all $t$, and it is such that $F^{*}_t \tilde{g}\mid_{f_g(M)}=g_t$ with $f_{g_t}: (Σ,g_t) \rightarrow (\mb{S}^{2+p},\tg)$ a path of minimal embedding deformations of the initial $f_g$. We apply this result to the Lawson surface $( Σ,g)=(ξ_{k/m,m}, g_{ξ_{k/m,m}})$, $m|k>1$, to conclude that if $a=μ_{g_{ξ_{k/m,m}}}(Σ)$, and $[0,1]\ni t \rightarrow g_t$ is a path of area $a$ metrics conformal deformations of $g_{k/m,m}$ to a metric $g_a$ of scalar curvature $4πχ(Σ)/a$, then $f_{g_{ξ_{k/m,m}}}: (ξ_{k/m,m},g_{ξ_{k/m,m}}) \rightarrow (\mb{S}^3, \tg)$ has associated minimal isometric conformal deformations $f_{g_t}$ to the isometric embedding $f_{g_a}$ of $g_a$, in sharp contrast with the situation of the standard sphere $ξ_{0,1}$ and Clifford torus $(ξ_{1,1}$, which are the only orientable Riemannian surfaces of genus $0$ and $1$ isometrically embedded into $(\mb{S}^3,\tg)$ as minimal surfaces. If $σ^2(Σ):=\sup_{[g]\in \mc{C}(Σ)}(4πχ( Σ))^2/\left(\frac{1}{4}\inf_{g\in [g]}\mc{W}_{f_g}(Σ)\right)$, $\mc{W}_{f_g}(Σ)$ the Willmore energy of $f_g$ and $\mc{C}( Σ)$ the space of classes, then $(4πχ(Σ))^2/\left( \frac{1} {4}\mc{W}_{f_g}(Σ) \right) \leq σ^2(Σ)=(4πχ( Σ))^2/\left(\frac{1}{4}\mc{W}_{f_{g_{ξ_{k,1}}}}(Σ) \right)$, and we describe the $f_g$s for which the equality is achieved.

math.DG↗

arXiv2026

Closed manifolds, model geometries, and volume related differentiable invariants

We view metrics through their isometric embeddigns $f_g:(M^n,g)\rightarrow (\mb{S}^{\tn},\tg)$ and their deformations. If $M$ carries a metric $g$ of constant scalar curvature $s_g$ and Ricci tensor $r_g \leq 0$, and if this $M$ does not carry scalar flat metrics other than Ricci flat ones, then $M$ is not a manifold of Kazdan-Warner (KW) type I, and if the space of Ricci flat metrics is not empty, $M$ is a manifold of KW type II, while otherwise, $M$ is of KW type III. If $M$ carries a metric $g'$ of nontrivial scalar curvature $s_{g'}\geq 0$, and an Einstein metric $g_{-}$ such that $r_{g_{-}}<0$, then $M$ must carry both, scalar flat non Ricci flat and Ricci flat metrics, and if orientable, it is spinnable. No such manifold exists if $n\leq 3$, and if $r_{g'}$ is assumed further to be positive, no such manifold exists if $n\leq 4$, and in these dimensions, $M^{n}$ can admit Einstein metrics of scalar curvature of at most one sign. If $M$ has a contractible universal cover and carries no Ricci flat metrics at all, $M$ is of KW type III. Based on these resulst, we find the KW type and sigma invariant of several manifolds $M=X/Γ_M$ with model geometry $({\rm Isom}(X,g),X)$ of Thurston. Notably, we show that an $M^n=\mb{H}^n/Γ_M$ of hyperbolic model $(\mb{H}^n,g_{\mb{H}^n})$ is of KW type III, that if $n\geq 3$ its $Γ_M$ invariant hyperbolic metric $g_M$ and class realize its sigma invariant, and that the space of hyperbolic metrics on $M$ is path connected and consists of isotopic deformations $f_{g_t}$ of $f_{g_0}:=f_{g_M}$ of equal volume metrics $g_t$ of constant sectional curvature $-1$, with $(M,g_t)$ isometric to $(M,g_M)$ for all $t$, while $3$d nil, solv, $\widetilde{\mb{P}\mb{S}\mb{L}}(2,\mb{R})$ and $\mb{R}\times \mb{H}^2$ manifolds are all of KW type III also, but have vanishing nonachievable sigma invariant.

math.DG↗

arXiv2025

Conformal properties of spheres

We identify the smooth metrics $\mc{M}(M)$ on a manifold $M^n$ with the smooth isometric embeddings $f_g: (M,g) \rightarrow (\mb{S}^{\tn}, \tg)$ into a standard sphere of large dimension $\tn=\tn(n)$, and their Palais isotopic deformations, and the space $\mc{C}(M)$ of conformal classes with the space of classes of metrics whose embeddings are isotopic to each other by conformal deformations. Isometric embeddings of a metric on the manifold with a different smooth structure, and their deformations, are carried by the same background also, but when they exist, they do not embed into a smooth flow of any $f_g(M)\hookrightarrow \mb{S}^{\tn}$. We characterize metrics of constant scalar curvature by properties of extrinsic quantities of their embeddings, and prove a homotopy lifting property of the bundle $\mc{M}(M) \stackrelπ{\rightarrow } \mc{C}(M)$ by Yamabe metrics, and when $M$ carries an almost complex structure $J_0$, extend it to a homotopy lifting property of the bundle $\mc{M}^{J_0}(M) \stackrelπ{\rightarrow } \mc{C}^{J_0}(M)$ of metrics compatible with almost complex structures in the same orientation class as $J_0$, and their conformal classes, the lift now by almost Hermitian Yamabe metrics. We use these results and the gap theorem of Simons to study the existence and integrability properties of almost complex structures on spheres, and products. We find the sigma invariants of $Sp(2)$, the $M^7_k$ spheres of Milnor, or any other, and except for $\mb{P}^1(\mb{C})\times \mb{P}^1(\mb{C})$, the almost Hermitian sigma invariant of product of spheres carrying almost complex structures, and organize manifolds with these invariants into a Pascal like triangle set according to the symmetries of the metrics, and the values of their associated conformal invariants.

math.DG↗

arXiv2025

Vector field cycles in the tangent bundle

Given a closed Riemannian manifold $(M^m,g)$ and a vector field $v$ on $M$, we form the Sasaki metric $g_S$ on $TM$, and restrict it to the image of the cross section map of $M$ into $TM$ defined by $v$, whose pull back to $M$ defines a new metric $g(v)$ on $M$. We then view the cross section as an isometric embedding $f_{g(v)}: (M,g(v))\rightarrow (TM,g_S)$, which when $\| v\|_g=1$, ranges into the unit sphere bundle $(S^1(TM),g_S)$. $v$ is minimal or minimal unit if these embeddings have null mean curvature vectors, conditions that occur if, $v$ is in the kernel or is an eigenvector, respectively, of a first order perturbation of a weighted rough Laplacian, the weights and perturbation determined by the covariant derivatives $\nabla^g_{e}v$ along unit directions $e$ in suitable normal frames that include $v$ when $\| v\|_g=1$, and curvature tensor of $g$. A minimal unit field must be Killing, and other than parallel fields, $v=0$ is the only minimal one. We characterize the minimal unit vector fields on the standard sphere $(\mb{S}^{2n+1},g) \hookrightarrow (\mb{R}^{2n+2},\| \, \|^2)$ as those defining contact strictly pseudoconvex CR structures whose Levi form and sign are determined by $g$ and the orientation. If $Θ_{f_{g(v)}}(M)$ and $Φ_{f_{g(v)}}(M)$ are the total exterior scalar curvature and squared $L^2$ norm of the mean curvature vector functionals, and $m>2$, a canonical cycle $f_{g(v)}(M)$ is a critical point of the functional $(m/m-1) Θ_{f_{g(v)}}(M) +Φ_{f_{g(v)}}(M)$ under conformal deformations, notion conveniently defined also when $m\leq 2$. The zero section of $TM$ is a canonical cycle if, and only if, the scalar curvature of $g$ is constant. We describe some examples of these vector fields and cycles, and analyze their deformations under dilations of the field.

math.DG↗

arXiv2024

On some manifolds with positive sigma invariants and their realizing conformal classes

We prove that the metric of the Riemannian product $(\mb{S}^k(r_1)\times \mb{S}^{n-k}(r_2), g^n_k)$, $r_1^2+r_2^2=1$, is a Yamabe metric in its conformal class if, and only if, either $g^n_k$ is Einstein, or the linear isometric embedding of this manifold into the standard $n+1$ dimensional sphere is minimal. We combine this result with Simons' gap theorem to show that, for $2\leq k\leq n-2$, the conformal class of the product metric with minimal embedding, which is at the upper end of Simons' gap theorem, realizes the sigma invariant of $\mb{S}^k\times \mb{S}^{n-k}$, and that this is the only class that achieves such a value. Similarly, we use coherent minimal isometric embeddings of suitably scaled standard Einstein metrics $g$ on $\mb{P}^n(\mb{R})$, $\mb{P}^n(\mb{C})$, and $\mb{P}^n(\mb{H})$ into unit spheres, and determine the sigma invariant of these projective spaces, prove that in each case the conformal class $[g]$ realizes it, and that this realizing class is unique.

math.DG↗

arXiv2023

Conformal invariants of isometric embeddings of the smooth metrics on a surface

We view all smooth metrics $g$ on a closed surface $Σ$ through their Nash isometric embeddings $f_g: (Σ,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We define the Willmore functional $\mathcal{W}_{f_g}$ over this space of metrics on $Σ$ in terms of the extrinsic quantities of $f_g$. Its infimum over metrics in a conformal class is an invariant of the class varying differentiably with it. If $Σ$ is oriented of genus $k$, when $k=0$, we use the gap theorem of Simons to show that there is a unique conformal class of metrics on $Σ$, whose invariant $16π$ is the value for the standard totally geodesic embedding of $\mathbb{S}^2 \hookrightarrow \mathbb{S}^{\tilde{n}}$, and we have that $\mathcal{W}_{f_g}(Σ) \geq 16π$, with the lower bound achieved if, and only if, $f_g$ is conformally equivalent to this standard geodesic embedding of $\mathbb{S}^2$, and ${\rm area}_g(Σ) \leq 4π$, while when $k\geq 1$, the Lawson minimal surface $(ξ_{k,1},g_{ξ_{k,1}})$ fixes the scale, and we show that $\mathcal{W}_{f_g}(Σ) \geq 4\, {\rm area}_{g_{ξ_{k,1}}} (ξ_{k,1})$, with the lower bound achieved by $f_g$ if, and only if, $f_g$ is conformally equivalent to $f_{g_{ξ_{k,1}}} : (ξ_{k,1}, g_{ξ_{k,1}}) \rightarrow (\mathbb{S}^3,\tilde{g}) \hookrightarrow (\mathbb{S}^{\tilde{n}},\tilde{g})$, and ${\rm area}_g(Σ)\leq {\rm area}_{g_{ξ_{k,1}}} (ξ_{k,1})$. For a nonoriented $Σ$, we prove a likewise estimate from below for $\mathcal{W}_{f_g}(Σ)$, and characterize conformally the surface that realizes the optimal lower bound.

math.DG↗

arXiv2018

The (small) vibrations of thin plates

We describe the equations of motion of elastodynamic bounded bodies in 3-space, and their linearizations at a stationary point. Using the latter as an approximation to model small motions, we develop a scheme to find numerical solutions of these equations. We discretize the solution in the space of PL vector fields associated to the oriented faces of the first barycentric subdivision of a given smooth initial triangulation of the body, in order to exploit the algebraic topology properties of the body that these vector fields encode into the sought after solution, and solve a weak version of the linearized equations in that context. We apply our scheme to a couple of relevant examples of thin bodies, bodies where one of the dimensions is at least one order of magnitude in size less than the other two, and determine numerical approximations to some of their resonance modes of vibration. The results obtained are consistent with known vibration patterns for these bodies derived experimentally.

math.NA↗

arXiv2014

The $L^2$-norm of the second fundamental form of isometric immersions into a Riemannian manifold

We consider critical points of the global squared $L^2$-norms of the second fundamental form and the mean curvature vector of isometric immersions into a fixed background Riemannian manifold under deformations of the immersion. We use the critical points of the former functional to define canonical representatives of a given integer homology class of the background manifold. We study the fibration ${\mathbb S}^3 \hookrightarrow Sp(2)\stackrel {π_{\circ}}{\rightarrow} {\mathbb S}^7$ from this point of view, showing that the fibers are the canonical generators of the $3$-integer homology of $Sp(2)$ when this Lie group is endowed with a suitable family of left invariant metrics. Complex subvarieties in the standard $\mb{P}^n(\mb{C})$ are critical points of each of the functionals, and are canonical representatives of their homology classes. We use this result to provide a proof of Kronheimer-Wrowka's theorem on the smallest genus representatives of the homology class of a curve of degree $d$ in ${\mathbb C}{\mathbb P}^2$, and analyze also the canonical representability of certain homology classes in the product of standard $2$-spheres. Finally, we provide examples of background manifolds admitting isotopically equivalent critical points in codimension one for the difference of the two functionals mentioned, of different critical values, which are Riemannian analogs of alternatives to compactification theories that has been offered recently.

math.DG↗

arXiv2013

Low energy canonical immersions into hyperbolic manifolds and standard spheres

We consider critical points of the functionals $Π$ and $Ψ$ defined as the global $L^2$-norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion. We prove gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the sphere can be cast as a theorem about critical points of these functionals of constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various type of minimal submanifolds that can occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of $Π$.

math.DG↗

arXiv2012

Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

Let (X,Ω) be a closed polarized complex manifold, g be an extremal metric on X that represents the Kähler class Ω, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deformations of (X,Ω)\simeq (M_0,Θ_0) provided with a holomorphic action of G with trivial action on B. Then for every t\in B sufficiently small, there exists an h^{1,1}(X)-dimensional family of extremal Kaehler metrics on M_t whose Kähler classes are arbitrarily close to Θ_t. We apply this deformation theory to show that certain complex deformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.

math.DG↗

arXiv2008

Arithmetic Laplacians

We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one dimensional groups, and analyze their spaces of solutions.

math.NT↗

arXiv2008

Arithmetic partial differential equations, II: modular curves

We classify ``arithmetic convection equations'' on modular curves, and describe their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions of the same modular forms; in this sense, our arithmetic convection equations can be seen as "unifying" the two types of expansions. The theory can be generalized to one of ``arithmetic heat equations'' on modular curves, but we prove that modular curves do not carry ``arithmetic wave equations.'' Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.

math.NT↗

arXiv2007

The Sasaki Cone and Extremal Sasakian Metrics

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest $L^2$-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension five, whose Sasaki cone coincides with the extremal set, and examples where the extremal set is empty. We end up by proving that a conjecture of Orlik concerning the torsion of the homology groups of these links holds in the five dimensional case.

math.DG↗

arXiv2007

Canonical Sasakian Metrics

Let $M$ be a closed manifold of Sasaki type. A polarization of $M$ is defined by a Reeb vector field, and for one such, we consider the set of all Sasakian metrics compatible with it. On this space, we study the functional given by the squared $L^2$-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open.

math.DG↗

arXiv2006

Arithmetic partial differential equations

We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to ``flow'' integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical ``flows'' on them that are the arithmetic analogues of the heat and wave equations. The same is true for the additive and the multiplicative group.

math.AP↗

arXiv2005

Heat Flows for Extremal Kähler Metrics

Let $(M,J,Ω)$ be a closed polarized complex manifold of Kähler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of Kähler metrics that are invariant under $G$ and represent the cohomology class $Ω$, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the $L^2$-norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its only fixed points are extremal metrics. We prove local time existence of the flow, and conclude that if the lifespan of the solution is finite, then the supremum of the norm of its curvature tensor must blow-up as time approaches it. While doing this, we also prove that extremal solitons can only exist in the non-compact case, and that the range of the holomorphy potential of the scalar curvature is an interval independent of the metric chosen to represent $Ω$. We end up with some conjectures concerning the plausible existence and convergence of global solutions under suitable geometric conditions.

math.DG↗