In Jain cosmology, time is divided into vast cycles called Kālas, each consisting of Utsarpiṇī (ascending half-cycle) and Avasarpiṇī (descending half-cycle). In each half-cycle, 63 illustrious persons known as Śalākā Puruṣas appear: 24 Tīrthaṅkaras, 12 Chakravartis (universal monarchs), 9 Balabhadras, 9 Narayanas, and 9 Pratinarayanas.

You are asking about the next Chakravarti from the Utsarpiṇī Kāla (the ascending half-cycle). According to Jain scriptures:

• In the current Avasarpiṇī (descending half-cycle), the last Chakravarti was King Bharata, son of the first Tīrthaṅkara, Lord Rishabhanatha.
• In the upcoming Utsarpiṇī (ascending half-cycle), the names of the 12 Chakravartis are also given in Jain texts.

The first Chakravarti of Utsarpiṇī Kāla will be King Swayambhu (or Swayambhu Manu in some traditions). The sequence and names of Chakravartis in Utsarpiṇī vary slightly between Jain traditions (Śvetāmbara and Digambara), but Swayambhu is generally recognized as the first Chakravarti in the ascending cycle.

So, the next (first) Chakravarti of Utsarpiṇī Kāla will be Swayambhu.

If you wish to know about the full list of 12 Utsarpiṇī Chakravartis, please let me know!