In this paper we will detail an approach to generate metrics and matter models on end-periodic manifolds, which are used extensively in the study of the exotic smooth structures of $\mathbb{R}^4$. We will present three distinct examples, discuss their associated matter models by solving the Einstein equations, and determine their physical viability by examining the energy conditions. We will also compare one of the models directly with existing models of matter distributions in extragalactic systems, to highlight the viability of utilizing exotic smooth structures to understand the existence and distribution of dark matter.
Equatorial periodic orbits and gravitational wave signatures in Euler-Heisenberg black holes surrounded by perfect fluid dark matter
Madhabdev University（马哈德夫大学）；Khazar University（阿塞拜疆哈萨尔大学）；Pacif Institute of Cosmology and Selfology (PICS)（太平洋宇宙学与自我学研究所（PICS））；Eastern Mediterranean University（东地中海大学）
Periodic Orbits and Their Gravitational-Wave Signatures around a Rotating Black Hole Surrounded by Perfect Fluid Dark Matter
Zhejiang University of Technology（浙江工业大学）；United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of Technology（引力波物理联合中心（UCGWP），浙江工业大学）；Bahauddin Zakariya University, Vehari Campus（巴哈丁·扎卡里亚大学，韦哈里校区）；Khazar University（哈扎尔大学）
Gravitational wave signatures and periodic orbits of a charged black hole in a Hernquist dark matter halo
Universidade Federal de Campina Grande（坎皮纳格兰德联邦大学）；Khazar University（阿塞拜疆可萨大学）；Damghan University（达姆甘大学）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Federal University of Paraíba（帕拉伊巴联邦大学）
