## Duke Mathematical Journal

Published by Duke University Press since its inception in 1935, the Duke Mathematical Journal is one of the world's leading mathematical journals. DMJ emphasizes the most active and influential areas of current mathematics. Advance publication of articles online is available.

On the elliptic equation $\Delta u+Ku^{(n+2)/(n-2)}=0$ and related topicsVolume 52, Number 2 (1985)
Linear differential equations on the Riemann sphere and representations of quiversVolume 166, Number 5 (2017)
The Möbius function and distal flowsVolume 164, Number 7 (2015)
Sur les représentations modulaires de degré $2$ de $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$Volume 54, Number 1 (1987)
A categorification of the Jones polynomialVolume 101, Number 3 (2000)
• ISSN: 0012-7094 (print), 1547-7398 (electronic)
• Publisher: Duke University Press
• Discipline(s): Mathematics
• Full text available in Euclid: 1935--
• Access: By subscription only
• Euclid URL: http://projecteuclid.org/dmj

### In memoriam

The Duke Mathematical Journal announces with sadness the passing of Duke Emeritus Professor Morris Weisfeld on 8 April 2017. Morris was Managing Editor of DMJ for all but 18 months of the period 1973–1997, during which time he increased its annual size from 1,000 to 3,000 pages. Under his leadership, DMJ became a much more interesting journal, with an eclectic mix of papers that contrasted (at the time) with Annals and Inventiones. More significantly, it was through his commitment to excellence that DMJ became one of the world’s most highly regarded mathematical journals.

Jonathan Wahl
Managing Editor, DMJ

### Featured bibliometrics

MR Citation Database MCQ (2015): 2.29
JCR (2015) Impact Factor: 2.350
JCR (2015) Five-year Impact Factor: 2.337
JCR (2015) Ranking: 9/312 (Mathematics)
Article Influence: 3.899
Eigenfactor: Duke Mathematical Journal
SJR/SCImago Journal Rank (2015): 5.675

Indexed/Abstracted in: Current Contents: Physical, Chemical and Earth Sciences, IBZ Online, Magazines for Libraries, MathSciNet, Science Citation Index, Science Citation Index Expanded, Scopus, and zbMATH

### Featured article

#### Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into $\operatorname{PSL}(3,\mathbb{R})$

Volume 166, Number 7 (2017)
##### Abstract

This article studies the geometry of proper open convex domains in the projective space $\mathbb{R}\mathbf{P}^{n}$. These domains carry several projective invariant distances, among which are the Hilbert distance $d^{H}$ and the Blaschke distance $d^{B}$. We prove a thin inequality between those distances: for any two points $x$ and $y$ in such a domain,

$$d^{B}(x,y)\lt d^{H}(x,y)+1.$$

We then give two interesting consequences. The first one answers a conjecture of Colbois and Verovic on the volume entropy of Hilbert geometries: for any proper open convex domain in $\mathbb{R}\mathbf{P}^{n}$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation $\rho$ of a surface group $\Gamma$ into $\operatorname{PSL}(3,\mathbb{R})$, there exists a Fuchsian representation $j:\Gamma\to\operatorname{PSL}(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than that of $\rho$. This answers positively a conjecture of Lee and Zhang in the $3$-dimensional case.