Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

Abstract

The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works.

Geometry and evolution of Hasimoto surface in Minkowski 3-space

“…In future work, we plan to investigate the harmonic evolute surfaces of the Hasimoto surface in different spaces, including Galilean and pseudo-Galilean spaces. We aim to enhance the results presented in this paper by incorporating techniques and findings from related studies [27][28][29][30][31][32][33][34][35][36][37]. Additionally, we intend to explore novel methods to discover further results and theorems concerning the singularity and symmetry properties of this topic, which will be presented in our upcoming papers.…”

Quasi-Statistical Schouten–van Kampen Connections on the Tangent Bundle

“…The new structures are, thus, uncorrelated with the ones from the base, therefore constituting a more convenient geometric setting to investigate the statistical behavior in depth. Thus, new opportunities are opened for applications in information theory, machine learning, neural networks, statistical mechanics and geometry of Ricci solitons, for which we cite [38][39][40][41] and the references therein.…”

Impact of Semi-Symmetric Metric Connection on Homology of Warped Product Pointwise Semi-Slant Submanifolds of an Odd-Dimensional Sphere

“…However, in [24], they also obtained some more results on warped product manifolds with a semi-symmetric metric connection. Furthermore, the studies mentioned in [25][26][27][28][29][30][31][32][33][34][35][36][37][38] are important contributions to soliton theory and submanifold theory, etc., related to the relevant topics. Motivated by these studies, we are interested in determining the impact of a semi-symmetric metric connection on the warped product pointwise semi-slant submanifolds and their homology in an odd-dimensional sphere.…”