Why only Lord Vishnu incarnates on earth?

Question

It is said that there are 3 supreme lords i.e. Brahma, Vishnu and Shiva. But from all 3 why only 10 incarnations of Lord Vishnu and not others?

Answer 1

No. Shiva and Brahma also incarnated on earth. It is indeed Shiva who takes Yogavatar at end of every Dwapara Yuga and beginning of Kali Yuga.

Most famous Avatar of Brahma is Chandra or Moon God which is explained in this answer. As explained in same answer, Sri Vaishnavas believe Vikhanasa as avatar of Brahma as described in Pancharatra Agamas.

Shiva's incarnations are described in detail in 42 chapters of Satarudra Samhita of Shiva Mahapurana. Some of His famous incarnations are Kala Bhairava, Veerabhadra, Rshba, Nandi (partial Avatar), Hanuman, Sage Pippalada, Drona's son Aswatthama. You can read their legends in detail from Satarudra Samhita which is available here (Chapter1, Chapter2 and Chapter3) and here (Chapter 80- Chapter 121).

Apart from Trimurties, Devas do take Avatars. As mentioned in this answer, many characters in Mahabharata and Ramayana are avatars of Devas like Arjuna being Nara and Indra, Krishna being sage Narayana (who is inturn Lord Vishnu), Karna being Surya, Bhisma being Vasu, Yudistara being Yama, Bheema being Vayudeva and so on. Number of Avatars of Devas with Avatars of Trimurties are infinite.

So, not only Lord Vishnu, other Devas too incarnated on earth.

Answer 2

Other devas incarnated also along with Vishnu for the welfare of the three worlds.

https://www.sacred-texts.com/hin/m01/m01065.htm

And, O bull of the Bharata race, when such was the blessed state of the terrestrial world, the Asuras, O lord of men, began to be born in kingly lines. And the sons of Diti (Daityas) being repeatedly defeated in war by the sons of Aditi (celestials) and deprived also of sovereignty and heaven, began to be incarnated on the earth. And, O king, the Asuras being possessed of great powers, and desirous of sovereignty began to be born on earth amongst various creatures, such as kine, horses, asses, camels, buffaloes, among creatures such as Rakshasas and others, and among elephants and deer. And, O protector of the earth, owing to those already born and to those that were being born, the earth became incapable of supporting herself. And amongst the sons of Diti and of Danu, cast out of heaven, some were born on the earth as kings of great pride and insolence.

And the Creator then commanded all the gods saying, 'To ease the Earth of her burden, go ye and have your births in her according to your respective parts and seek ye strife (with the Asuras already born there)'. And the Creator of all, summoning also all the tribes of the Gandharvas and the Apsaras, spake unto them these words of deep import, 'Go ye and be born amongst men according to your respective parts in forms that ye like.' So the gods incarnated as most important characters of Mahabharata to ease the burden

https://www.sacred-texts.com/hin/m01/m01066.htm

"And the dwellers in heaven gradually became incarnate on earth for the destruction of the Asuras and for the welfare of the three worlds."

https://www.sacred-texts.com/hin/m01/m01068.htm

And from the curse of Vasishtha and the command also of Indra, the eight Vasus were born of Ganga by her husband Santanu. The youngest of them was Bhishma, the dispeller of the fears of the Kurus, gifted with great intelligence, conversant with the Vedas, the first speakers, and the thinner of the enemy's ranks.

And, O monarch, know thou that Drona, the son of Bharadwaja, not born of any woman, sprung from a portion of the celestial Rishi Brihaspati of grand achievements.

And, O king, his son the heroic Ashwatthama, of eyes like the lotus-petals, gifted with surpassing energy, and the terror of all foes, the great oppressor of all enemies, was born on earth, of the united portions of Mahadeva, Yama, Kama, and Krodha.

Kripa was the embodiment of all manliness was born of the tribe of the Rudras.

Sakuni, that crusher of foes, was Dwapara himself (the third yuga).

And he who was Satyaki of sure aim, that upholder of the pride of Vrishni race, that oppressor of foes, begotten of the portion of gods called the Maruts. And that royal sage Drupada who on earth was a monarch, the first among all persons bearing arms, was also born of the same tribe of the celestials. And, O king, thou shouldst also know that Kritavarma, that prince among men, of deeds unsurpassed by any one, and the foremost of all bulls amongst Kshatriyas, was born of the portion of the same celestials. And that royal sage also, Virata by name, the scorcher of the kingdoms of others, and the great oppressor of all foes, was born of the portion of the same gods.

Pandu, devoted to truth and virtue, was Purity's self

Vidura, who was the first of all virtuous men, who was the god of Justice himself

The evil-minded and wicked king Duryodhana, the destroyer of the fair fame of the Kurus, was born of a portion of Kali on earth.

the sons of Pulastya (the Rakshasas) were born on earth among men of Duryodhana's brothers - other remaining 99 kauravas besides Duryodhana

king Yudhishthira was a portion of Dharma; that Bhimasena was of the deity of wind; that Arjuna was of Indra, the chief of the celestials; and that Nakula and Sahadeva, the handsomest beings among all creatures, and unrivalled for beauty on earth, were similarly portions of the twin Aswins.

Varchas, the son of Soma, became Abhimanyu of wonderful deeds, the son of Arjuna.

Dhrishtadyumna was a portion of Agni.

Sikhandin, who was at first a female, was (the incarnation of) a Rakshasa.

five sons of Draupadi, those bulls amongst the Bharata princes, were the celestials known as the Viswas.

Draupadi, slender-waisted like the wasp, was born of a portion of Sachi (the queen of the celestials), in the line of Drupada.

two goddesses Siddhi and Dhriti became the mothers of those five, and were called Kunti and Madri.

And a portion of Sri herself became incarnate on earth, for the gratification of Narayana, in the line of Bhishmaka. And she was by name the chaste Rukmini.

Mati became the daughter (Gandhari) of Suvala.

Valadeva of exceeding strength was a portion of the Naga, Sesha.

Pradyumna of great energy was Sanatkumara.

Karna--the first of all exalted men--the foremost of all wielders of weapons--the slayer of foes--and the best portion of the maker of day--was the friend and counsellor of Duryodhana.

the portions of the tribe of Apsaras - sixteen thousand portions of those goddesses became, O king, in this world of men, the wives of Vasudeva.

And he, called Vasudeva, endued with great valour, was among men a portion of him called Narayana--the god of gods--eternal.

Ghatotkacha - Indeed, it was the illustrious Indra who created (by lending a portion of himself) the mighty car-warrior Ghatotkacha as a fit antagonist of Karna of unrivalled energy, in consequence of the dart he had given unto Karna (and which was sure to kill the person against whom it would be hurled).'"

Not only Lord Vishnu but many gods also aided him in the divine task which was to ease the burden on earth.