arXiv subjects

Shubham Dwivedi

Publications and source records attributed to Shubham Dwivedi.

11 recordsLinked to original sources

arXiv2026

A $dd^{\Phi}$-Lemma and Bott--Chern-type Cohomology for Spin(7)-Manifolds

We study the properties of the $dd^{\Phi}$-operator on $8$-dimensional Spin(7)-manifolds with torsion-free Spin(7)-structures $\Phi$. These operators were first introduced by Harvey and Lawson (An introduction to potential theory in calibrated geometry. Am.J. Math. 131.4 (2009), arXiv:0710.3920). We prove a Hodge decomposition theorem for the $dd^{\Phi}$-operator and obtain an analogue of the $\partial \bar{\partial}$-lemma in K\"ahler geometry. Using this, we define Bott--Chern-type cohomologies for Spin(7)-manifolds. We relate the Bott-Chern-type cohomology spaces to the moduli space of torsion-free Spin(7)-structures and calibrated geometry of Spin(7)-manifolds. These relations naturally give rise to the notion of Cayley-positive cones. In the course of proving the results, we state and prove various identities for the exterior derivative and its decompositions into irreducible Spin(7)-representations as well as identities for second order derivatives and Laplacians. The identities we prove are for any Spin(7)-structures and the specialized torsion-free ones are Spin(7)-analogoues of K\"ahler identities and Bryant--Harvey's identities in the $\mathrm{G}_2$-case (R. Bryant, Some remarks on $\mathrm{G}_2$-structures, Proceedings of the 11th and 12th G\"okova geometry-topology conference, arXiv:math/0305124) and are results of independent interest.

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arXiv2026

Solutions and singularities of the Ricci-harmonic flow and Ricci-like flows of $\mathrm{G_2}$-structures

We find explicit solutions and singularities of the Ricci-harmonic flow of $\mathrm{G_2}$-structures, the Ricci-like flows of $\mathrm{G_2}$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and of the negative gradient flow of an energy functional of $\mathrm{G_2}$-structures, on $7$-dimensional contact Calabi-Yau manifolds and the $7$-dimensional Heisenberg group. We prove that the natural co-closed $\mathrm{G_2}$-structure on a contact Calabi-Yau manifold as the initial condition leads to an ancient solution of the Ricci-harmonic flow with a finite time Type I singularity, and it gives an immortal solution to the Ricci-like flows with an infinite time singularity which are Type III if the transversal Calabi-Yau distribution is flat, and Type IIb otherwise. The same ansatz gives ancient solution to the negative gradient flow of $\mathrm{G_2}$-structures. These are the first examples of Type I singularities of the Ricci-harmonic flow and Type IIb and Type III singularities of the Ricci-like flows. We also obtain similar solutions for all the three flows on the $7$-dimensional Heisenberg group.

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arXiv2026

Ricci-harmonic flow of $\mathrm{G}_2$ and Spin(7)-structures

We introduce and study a new general flow of $\mathrm{G}_2$-structures which we call the Ricci-harmonic flow of $\mathrm{G}_2$-structures. The flow is the coupling of the Ricci flow of underlying metrics and the isometric flow of $\mathrm{G}_2$-structures, but we also provide explicit lower order in the torsion terms. The lower order terms and the flow are obtained by analyzing the second order term in the Taylor series expansion of $\mathrm{G}_2$-structures in normal coordinates. As such, the Ricci-harmonic flow described in the paper can be interpreted as the "heat equation" for $\mathrm{G}_2$-structures. The lower order terms allow us to prove that the stationary points of the Ricci-harmonic flow are exactly torsion-free $\mathrm{G}_2$-structures on compact manifolds. We study various analytic and geometric properties of the flow. We show that the flow has short-time existence and uniqueness on compact manifolds starting with an arbitrary $\mathrm{G}_2$-structure and prove global Shi-type estimates. We also prove a modified local Shi-type estimates for the flow which assume bounds on the initial derivatives of the Riemann curvature tensor and the torsion but give uniform bounds on these quantities for all times. We prove a compactness theorem for the solutions of the flow and use it to prove that the Ricci-harmonic flow exists as long as the velocity of the flow remains bounded. We also study Ricci-harmonic solitons where we prove that there are no compact expanding solitons and the only steady solitons are torsion-free. We derive an analog of Hamilton's identity for gradient Ricci-harmonic solitons and prove some integral identities for the solitons. Finally, we prove a version of the Taylor series expansion for Spin(7)-structures and use it to derive the Ricci-harmonic flow of Spin(7)-structures.

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arXiv2026

A gradient flow of Spin(7)-structures

We formulate and study the negative gradient flow of an energy functional of Spin(7)-structures on compact $8$-manifolds. The energy functional is the $L^2$-norm of the torsion of the Spin(7)-structure. Our main result is the short-time existence and uniqueness of solutions to the flow. We also explain how this negative gradient flow contains, as the highest order terms, all independent second order differential invariants of Spin(7)-structures which can be made into an admissible $4$-form. We also study solitons of the flow and prove a non-existence result for compact expanding solitons.

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arXiv2025

Flows of $G_2$-structures, II: Curvature, torsion, symbols, and functionals

We continue the investigation of general geometric flows of $G_2$-structures initiated by the third author in "Flows of $G_2$-structures, I." Specifically, we determine the possible geometric flows (up to lower order terms) of $G_2$-structures which are second order quasilinear, by explicitly computing all independent second order differential invariants of $G_2$-structures which are $3$-forms. There are four symmetric $2$-tensors and two vector fields. We do this by deriving explicit computational descriptions of the decompositions of the curvature and the covariant derivative of the torsion into irreducible $G_2$-representations, as well as the decomposition of the $G_2$-Bianchi identity into independent relations. We also show that these six tensors arise as leading order contributions to the Euler-Lagrange equations for the energy functionals of the four independent torsion components, and we establish a $G_2$-analogue of the classical block decomposition of the Riemann curvature operator on oriented $4$-dimensional Riemannian manifolds. Finally, we present a large class of geometric flows of $G_2$-structures which are directly amenable to a deTurck type trick to establish short-time existence and uniqueness, with no initial assumption on the torsion, vastly generalizing an earlier result of Weiss-Witt for the negative gradient flow of the Dirichlet energy. This result is proved through a careful analysis of the principal symbols of the linearizations of these operators, establishing particular linear combinations for which one can prove that the failure of strict parabolicity is due precisely to the diffeomorphism invariance. A detailed introductory section on various foundational results of $G_2$-structure, several of which are not readily available in the literature, should be of wider interest and applicability.

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arXiv2022

Associative submanifolds in Joyce's generalised Kummer constructions

This article constructs examples of associative submanifolds in $G_2$-manifolds obtained by resolving $G_2$-orbifolds using Joyce's generalised Kummer construction. As the $G_2$-manifolds approach the $G_2$-orbifolds, the volume of the associative submanifolds tends to zero. This partially verifies a prediction due to Halverson and Morrison.

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arXiv2021

Harmonic flow of $\mathrm{Spin}(7)$-structures

We formulate and study the isometric flow of $\mathrm{Spin}(7)$-structures on compact $8$-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of $H$-structures. We then specialise to $H=\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an $\varepsilon$-regularity theorem. Moreover, we prove Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-$\mathrm{I}$ singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric $\mathrm{Spin}(7)$-structures, based on squares of spinors, which may be of independent interest.

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arXiv2020

Deformation theory of nearly $\mathrm{G}_2$ manifolds

We study the deformation theory of nearly $\mathrm{G}_2$ manifolds. These are seven dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $\mathrm{G}_2$ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $\mathrm{G}_2$ structure on the Aloff--Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly $\mathrm{G}_2$ manifolds.

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arXiv2020

Some results on Ricci-Bourguignon solitons and almost solitons

We prove some results for the solitons of the Ricci-Bourguignon flow, generalizing corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon $almost$ solitons and prove some results about them which generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci-Bourguignon solitons and compact gradient Ricci-Bourguignon almost solitons. Finally, using the integral formula we show that a compact gradient Ricci-Bourguignon almost soliton is isometric to an Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

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arXiv2019

A gradient flow of isometric $\mathrm{G}_2$ structures

We study a flow of $G_2$ structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time singularity the torsion must blow-up, so the flow exists as long as the torsion remains bounded. We prove a Cheeger-Gromov type compactness theorem for the flow. We describe an Uhlenbeck-type trick which together with a modification of the connection gives a nice diffusion-reaction equation for the torsion along the flow. We define a quantity for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi on the mean curvature flow, we define an entropy functional and after proving an $ε$-regularity theorem, we show that low entropy initial data lead to solutions of the flow which exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We also study the finite-time singularities and show that at the singular time the flow converges to a smooth $G_2$ structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution has a subsequence which converges to a shrinking soliton for the flow.

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arXiv2018

Minimal Hypersurfaces in nearly $\mathrm{G}_2$ Manifolds

We study hypersurfaces in a nearly $\mathrm{G}_2$ manifold. We define various quantities associated to such a hypersurface using the $\mathrm{G}_2$ structure of the ambient manifold and prove several relationships between them. In particular, we give a necessary and sufficient condition for a hypersurface with an almost complex structure induced from the $\mathrm{G}_2$ structure of the ambient manifold, to be nearly K$\ddot{\text{a}}$hler. Then using the nearly $\mathrm{G}_2$ structure on the round sphere $S^7$, we prove that for a compact minimal hypersurface $M^6$ of constant scalar curvature in $S^7$ with the shape operator $A$ satisfying $|A|^2>6$, there exists an eigenvalue $λ>12$ of the Laplace operator on $M$ such that $|A|^2=λ- 6$, thus giving the next discrete value of $|A|^2$ greater than $0$ and $6$, thus generalizing an earlier result about nearly K$\ddot{\text{a}}$hler $S^6$.

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