We study a flow of G2-structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist 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 underlying connection yields a nice reaction-diffusion equation for the torsion along the flow. We define a scale-invariant quantity Θ for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi [CM12] 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 that exist for all time and converge smoothly to a G2-structure with divergence-free torsion. We also study finite-time singularities and show that at the singular time the flow converges to a smooth G2-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 admits a subsequence that converges to a shrinking soliton for the flow.

We prove some results for the solitons of the Ricci–Bourguignon flow, generalizing the 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 that 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 a Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

A robust preclinical disease model is a primary requirement to understand the underlying mechanisms, signaling pathways, and drug screening for human diseases. Although various preclinical models are available for several diseases, clinical models for Alzheimer’s disease (AD) remain underdeveloped and inaccurate. The pathophysiology of AD mainly includes the presence of amyloid plaques and neurofibrillary tangles (NFT). Furthermore, neuroinflammation and free radical generation also contribute to AD. Currently, there is a wide gap in scientific approaches to preventing AD progression. Most of the available drugs are limited to symptomatic relief and improve deteriorating cognitive functions. To mimic the pathogenesis of human AD, animal models like 3XTg-AD and 5XFAD are the primarily used mice models in AD therapeutics. Animal models for AD include intracerebroventricular-streptozotocin (ICV-STZ), amyloid beta-induced, colchicine-induced, etc., focusing on parameters such as cognitive decline and dementia. Unfortunately, the translational rate of the potential drug candidates in clinical trials is poor due to limitations in imitating human AD pathology in animal models. Therefore, the available preclinical models possess a gap in AD modeling. This paper presents an outline that critically assesses the applicability and limitations of the current approaches in disease modeling for AD. Also, we attempted to provide key suggestions for the best-fit model to evaluate potential therapies, which might improve therapy translation from preclinical studies to patients with AD.

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 is the most general flow of Spin(7)-structures. We also study solitons of the flow and prove a nonexistence result for compact expanding solitons.

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