arXiv subjects

Doanh Pham

Publications and source records attributed to Doanh Pham.

5 recordsLinked to original sources

arXiv2026

Determinant majorization for hyperbolic polynomials on Euclidean Jordan algebras

Under cone containment and the central ray hypothesis, we prove a determinant majorization result for hyperbolic polynomials on Euclidean Jordan algebras. In the case of real symmetric $n \times n$ matrices, this recovers the main theorem of Harvey and Lawson [Duke Math. J. 174 (2025), no. 13, 2749--2763] and also yields a new $\sigma_2$ majorization result. Moreover, for every $3 \leq k \leq n-1$, we construct explicit hyperbolic polynomials satisfying cone containment and the central ray hypothesis but for which the analogous $\sigma_k$ majorization fails.

math.AP↗

arXiv2026

A monotonicity formula for anisotropic minimal hypersurfaces

Under a sign assumption on the Minkowski norm, we prove a monotonicity formula for anisotropic minimal hypersurfaces in Euclidean space.

math.DG↗

arXiv2025

Monotonicity formulas for minimal submanifolds involving M\"obius transformations

For a minimal submanifold of the Euclidean space, we prove monotonicity formulas for its (weighted) volume within images of concentric balls under M\"obius transformations.

math.DG↗

arXiv2024

A logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector

In this paper, we prove a logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector in a manifold with nonnegative sectional curvature.

math.DG↗

arXiv2023

Some geometric inequalities by the ABP method

In this paper, we apply the so-called Alexandrov-Bakelman-Pucci (ABP) method to establish some geometric inequalities. We first prove a logarithmic Sobolev inequality for closed $n$-dimensional minimal submanifolds $Σ$ of $\mathbb S^{n+m}$. As a consequence, it recovers the classical result that $|\mathbb S^n| \leq |Σ|$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for positive symmetric two-tensors on smooth domains in $\mathbb R^n$ which was established by D. Serre when the domain is convex. In the last application of the ABP method, we formulate and prove an inequality related to quermassintegrals of closed hypersurfaces of the Euclidean space.

math.DG↗