## 1 Introduction  
As an indispensable part of the future 6G vision [1, 2], the ultra-dense low Earth orbit (LEO) satellite communication (UD-LSC) networks such as SpaceX and OneWeb can provide low latency, seamless coverage, and broadband communication [3]. However, the rapid construction of these constellations makes the competition between orbit resources and frequency resources in space distinctly fierce. Spectrum sharing between different communication systems is a promising approach to ease the spectrum environment [4], but the deployment of a large amount of LEO satellites brings in significant inter-system interference with the geostationary Earth orbit (GEO) satellite communication systems in service and intra-system interference between LEO satellites due to the dense topology. To track this problem, advanced interference mitigation technologies should be taken into consideration. Many efforts have been devoted to addressing the inter-system interference when spectrum sharing between the GEO and LEO communication systems is adopted. The cognitive radio (CR) technology [5] that senses the unused spectrum and takes opportunistic access is considered in the LEO satellite systems [6] to share the same frequency with GEO satellites, but the available resource of the secondary system is rigorously restricted. In [7], the interference effect from the LEO satellite communication system to the GEO satellite communication system is analyzed by BER performance. Simulation results show that the off-axis angle between the LEO satellite and the GEO satellite has a significant influence on interference. However, specific solutions for dealing with the inter-system interference are not included. In [8], the adaptive power control (APC) technique that adapts the transmit power of the NGEO satellite/terminal in order to guarantee that the interference to the GEO link is below the tolerable interference limit is proposed. However, only mitigating interference from the power domain will seriously affect the service performance of the NGEO system. In [9], the in-line interference was mitigated by adjusting the direction of phased array antennas of LEO satellites. In this paper, both the angular domain and power domain are taken into consideration to achieve spectrum sharing between the GEO and the UD-LSC networks. The above related research works focused on the in-line interference between low-dense LEO satellite constellations and the GEO satellite. With the dense topology of the UD-LSC networks, the intra-system interference in UD-LSC networks should be further considered. The intra-system interference is the interference between different beams of the same or adjacent LEO satellites. Generally, segmenting the spectrum in two or four colors with different polarization modes is a promising way to suppress the intra-system interference. However, the spectrum efficiency is correspondingly decreased. The on-board or on-ground precoding/beamforming with aggressive full frequency reuse (FFR) is researched in [10] to address the severe interference, and a frame-based multigroup multicast precoding scheme is proposed based on DVB-S2X protocol [11]. However, the computation of high-dimensional beamforming matrices is a challenging task. In the UD-LSC network, not only does the interference between the beams of one single LEO satellite need to be considered but also the interference between adjacent multiple LEO satellites. In this paper, spectrum sharing between the GEO satellite and the UD-LSC network is studied. For simplicity, this paper focuses on analyzing the spectrum sharing between GEO and UD-LSC networks, and it does not consider MEO and other systems. However, the research method can be extended to scenarios where there are other systems. The main contributions of this paper are summarized as follows. * We first propose a multi-satellite cooperative framework for UD-LSC networks to share the same spectrum with the GEO satellite, taking both the inter-system interference and intra-system interference into consideration.